353117 NOTRE DAME MATHEMATICAL LECTURES Number 2

GALOIS THEORY

Lectures delivered at the University of Notre Dame

by

DR. EMIL ARTIN Professor of Mathematics, Princeton University

Edited and supplemented with a Section on Applications

by

DR. ARTHUR N. MILGRAM Associate Professor of Mathematics, University of Minnesota

Second Edition With Additions and Revisions

UNIVERSITY OF NOTRE DAME PRESS NOTRE DAME LONDON Copyright 1942, 1944 UNIVERSITY OF NOTRE DAME

Second Printing, February 1964 Third Printing, July 1965 Fourth Printing, August 1966 New composition with corrections Fifth Printing, March 1970 Sixth Printing, January 197 1

Printed in the United States of America by NAPCO Graphie Arts, Inc., Milwaukee, Wisconsin TABLE OF CONTENTS

(The sections marked with an asterisk have been herein added to the content of the first edition)

Page 1 LINEAR ALGEBRA ...... 1 A. Fields...... 1 B. Vector Spaces ...... 1 C. Homogeneous Linear Equations ...... 2 D. Dependence and Independence of Vectors .. , ...... 4 E. Non-homogeneous Linear Equations ...... 9 F.* Determinants ...... 11 II FIELD THEORY ...... <...... 21 A. Extension Fields ...... 21 B. Polynomials ...... 22 C. Algebraic Elements ...... 25 D. Splitting Fields ...... 30 E. Unique Decomposition of Polynomials into Irreducible Factors ...... , ...... 33 F. Group Characters ...... 34 G.* Applications and Examples to Theorem 13 ...... 38 H. Normal Extensions ...... 41 Finite Fields ...... 49 J. Roots of Unity ...... , .. 56 K. Noether Equations ...... 57 L. Kummer’s Fields ...... 59 M. Simple Extensions ...... 64 N. Existence of a Normal Basis ...... , ...... 66 Q. Theorem on Natural Irrationalities ...... 67 111 APPLICATIONS By A. N. Milgram., ...... , ...... 69 A. Solvable Groups ...... 69 B. Permutation Groups ...... 70 C. Solution of Equations by Radicals ...... 72 D. The General Equation of Degree n...... 74 E. Solvable Equations of Prime Degree ...... 76 F. Ruler and Compass Construction ...... 80

1 LINEAR ALGEBRA

A. Fie’lds--* A field is a set of elements in which a pair of operations called multiplication and addition is defined analogous to the operations of multipl:ication and addition in the real number system (which is itself an example of a field). In each field F there exist unique elements called o and 1 which, under the operations of addition and multiplica- tion, behave with respect to a11 the other elements of F exactly as their correspondents in the real number system. In two respects, the analogy is not complete: 1) multiplication is not assumed to be commu- tative in every field, and 2) a field may have only a finite number of elements. More exactly, a field is a set of elements which, under the above mentioned operation of addition, forms an additive abelian group and for which the elements, exclusive of zero, form a multiplicative group and, finally, in which the two group operations are connected by the distributive law. Furthermore, the product of o and any element is de- fined to be o. If multiplication in the field is commutative, then the field is called a commutative field.

B. Vector Spaces. If V is an additive abelian group with elements A, B, . . . , F a field with elements a, b, . . . , and if for each a c F and A e V 2 the product aA denotes an element of V, then V is called a (left) vector space over F if the following assumptions hold: 1) a(A + B) = aA + aB 2) (a + b)A = aA + bA 3) a(bA) = (ab)A 4) 1A = A The reader may readily verify that if V is a vector space over F, then oA = 0 and a0 = 0 where o is the zero element of F and 0 that of V. For example, the first relation follows from the equations: aA = (a + o)A = aA + oA

Sometimes products between elements of F and V are written in the form Aa in which case V is called a right vector space over F to distinguish it from the previous case where multiplication by field ele- ments is from the left. If, in the discussion, left and right vector spaces do not occur simultaneously, we shall simply use the term “vector space.”

C. Homogeneous Linear Equations. If in a field F, aij, i = 1,2,. . . , m, j = 1,2, . . . , n are m . n ele- ments, it is frequently necessary to know conditions guaranteeing the existence of elements in F such that the following equations are satisfied: a,, xi + a,, x2 + . . . + alnxn = 0.

(1) . * a ml~l + amzx2 + . . . + amnxn = 0.

The reader Will recall that such equations are called linear homogeneous equations, and a set of elements, xi, x2,. . . , xr, of F, for which a11 the above equations are true, is called 3 a solution of the system. If not a11 of the elements xi, xg, . . . , xn are o the solution is called non-trivial; otherwise, it is called trivial.

THEOREM 1. A system of linear homogeneous equations always has a non-trivial solution if the number of unknowns exceeds the num- ber of equations. The proof of this follows the method familiar to most high school students, namely, successive elimination of unknowns. If no equations in n > 0 variables are prescribed, then our unknowns are unrestricted and we may set them a11 = 1. We shall proceed by complete induction. Let us suppose that each system of k equations in more than k unknowns has a non-trivial solution when k < m. In the system of equations (1) we assume that n > m, and denote the expression a,ixi + . . . + ainxn by L,, i = 1,2,. . .,m. We seek elements xi, . . . , x,, not a11 o such that L, = L, = . . . = Lm = o. If aij = o for each i and j, then any choice of xi , . . . , xr, Will serve as a solution. If not a11 aij are o, then we may assume that ail f o, for the order in which the equations are written or in which the unknowns are numbered has no influence on the existence or non-existence of a simultaneous solution. We cari find a non-trivial solution to our given system of equations, if and only if we cari find a non-trivial solution to the following system:

L, = 0

L, - a,,a,;lL, = 0

. . . . . Lm - amia,;lL, = 0 For, if xi,. . . , x,, is a solution of these latter equations then, since L, = o, the second term in each of the remaining equations is o and, hence, L, = L, = . . . = Lm = o. Conversely, if (1) is satisfied, then the new system is clearly satisfied. The reader Will notice that the new system was set up in such a way as to “eliminate” x1 from the last m-l equations. Furthermore, if a non-trivial solution of the last m-l equations, when viewed as equations in x2, . . . , xn, exists then taking xi = - ai;‘( ai2xz + ar3x3 + . . . + alnxn) would give us a solution to the whole system. However, the last m-l equations have a solution by our inductive assumption, from which the theorem follows. Remark: If the linear homogeneous equations had been written in the form xxjaij = o, j = 1,2, . . . , n, the above theorem would still hold and with the same proof although with the order in which terms are written changed in a few instances.

D. Dependence and Independence of Vectors. In a vector space V over a field F, the vectors A,, . . . , An are called dependent if there exist elements xi, . . . , x”, not a11 o, of F such that xiA, + x2A, + . . . + xnAn = 0. If the vectors A,, . . . ,An are not dependent, they are called independent.

The dimension of a vector space V over a field F is the maximum number of independent elements in V. Thus, the dimension of V is n if there are n independent elements in V, but no set of more than n independent elements.

A system A,, . . . , A, of elements in V is called a generating system of V if each element A of V cari be expressed 5 linearly in terms of A,, . . . , Am, i.e., A = Ca.A. for a suitable choice i=ll 1 ofa,, i = l,..., m,inF.

THEOREM 2. In any generating system the maximum number of independent vectors is equal to the dimension of the vector space.

Let A,, . . . , A,,, be a generating system of a vector space V of dimension n. Let r be the maximum number of independent elements in the generating system. By a suitable reordering of the generators we may as- sumek,, . . . , Ar independent. By the definition of dimension it follows that r -< n. For each j, A,, . . . , A,. A,+j are dependent, and in the relation

a,A, + a,A, + ..* + arAr + a,+j A,+j = 0 expressing this, a ,+j # o, for the contrary would assert the dependence of A,, . . . ,Ar. Thus,

A,+j = - ar+y[a,A, + a,A, + . . . + arAr].

It follows that A,, . . . , Ar is also a generating system since in the linear relation for any element of V the terms involving Ar+j, j f o, cari a11 be replaced by linear expressions in A,, . . . , Ar.

Now, let B,, . . . , B, be any system of vectors in V where t > r, then there exist aij such that Bj =iglaij Ai, j = 1,2, . . . , t, since the Ai’ s form a generating system. If we cari show that B,, . . . , B, are dependent, this Will give us r -> n, and the theorem Will follow from this together with the previous inequality r <- n. Thus, we must ex- hibit the existence of a non-trivial solution out of F of the equation

xiB, + x2B, + . . . + xrB, = 0. 6

TO this end, it Will be sufficient to choose the xi’s SO as to satisfy the linear equationsiir xj aij = o, i = 1,2,. . . , r, since these ex- pressions Will be the coefficients of Ai when in E x. B. the Bj ‘s are r j=l J J replaced by 2 aij Ai and terms are collected. A solution to the equa- i=l tions 2 xjaij = 0, i = 1,2,. . . , r, always exists by Theorem 1. .i=l Remark: Any n independent vectors A,, . . . , A,, in an n dimen- sional vector space form a generating system. For any vector A, the vectors A, A,, . . . , A,, are dependent and the coefficient of A, in the dependence relation, cannot be zero. Solving for A in terms of

A l>“‘> A,, exhibits A,, . . . ,An as a generating system.

A subset of a vector space is called a subspace if it is a sub- group of the vector space and if, in addition, the multiplication of any element in the subset by any element of the field is also in the subset. If Ai,..., AS are elements of a vector space V, then the set of a11 ele- ments of the form a, A, + . . . + asAS clearly forms a subspace of V. It is also evident, from the definition of dimension, that the dimension of any subspace never exceeds the dimension of the whole vector space.

An s-tuple of elements ( a,, . . . , as ) in a field F Will be called a --row vector. The totality of such s-tuples form a vector space if we define

a) (a,,a, ,..., as) = (b,,b, ,..., bS)ifandonlyif

a, = b,, i = 1,. . . , s,

B> (alta2,...,as) + (bl,b2,...,bs) = (a1 + b,,a, + b,,

. . ..aS + bs), y) b(a,,a, ,..., as) = (ba,,ba, ,..., baS),forban element of F. When the s-tuples are written vertically,

they Will be called column vectors. THEOREM 3. The row (column) vector space F” of a11 n-tuples from a field F is a vector space of dimension n over F. The n elements Cl = (l,o,o ,...> 0) E2 = (o,l,o >..., 0)

6, = (o,o,...,o,l) are independent and generate F”. Both remarks follow from the relation (a1,a2,. . . ,an) = Xaici. We cal1 a rectangular array

of elements of a field F a matrix. By the right row rank of a matrix, we mean the maximum number of independent row vectors among the rows (ail,..., a,,) of the matrix when multiplication by field elements is from the right. Similarly, we define left row rank, right column rank and left column rank. THEOREM 4. In any matrix the right column rank equals the left row rank and the left column rank equals the right row rank. If the field 8 is commutative, these four numbers are equal to each other and are called the rank of the matrix. Cal1 the column vectors of the matrix C,, . ~ . , Cn and the row vectors R,, . . . , Rm. The column vector 0 is o and any 0

(:)0 dependence Crx, + C,x, + . . . + Cnx, = 0 is equivalent to a solution of the equations

arrxr + a12x2 + . . . + a lnxn = O

(1) :

amlxl + amZx2 + . . . + a nmxn = 0. Any change in the order in which the rows of the matrix are written gives rise to the same system of equations and, hence, does not change the column rank of the matrix, but also does not change the row rank since the changed matrix would have the same set of row vectors. Cal1 c the right column rank and r the left row rank of the matrix. By the above remarks we may assume that the first r rows are independent row vectors. The row vector space generated by a11 the rows of the matrix has, by Theorem 1, the dimension r and is even generated by the first r rows. Thus, each row after the rth .1s linearly expressible in terms of the first r rows. Consequently, any solution of the first r equations in (1) Will be a solution of the entire system since any of the last n-r equations is obtainable as a linear combination of the first r. Con- versely, any solution of (1) Will also be a solution of the first r equations. This means that the matrix 9

%la12- * .%n ...... arr ar2. . . a rn consisting of the first r rows( of the original4 matrix has the same right column rank as the original. It has also the same left row rank since the r rows were chosen independent. But the column rank of the ampu- tated matrix car-mot exceed r by Theorem 3. Hence, c <- r. Similarly, calling c’ the left column rank and r’ the right row rank, c’ <- r’. If we form the transpose of the original matrix, that is, replace rows by columns and columns by rows, then the left row rank of the transposed matrix equals the left column rank of the original. If then to the transposed matrix we apply the above considerations we arrive at r -< c and r’ <-. c’.

E. -Non-homogeneous Linear Equations. The system of non-homogeneous linear equations

arrxi + ar2x2 + . . . + alnxn = bl azlxl + ...... + aznxn = b2 (2) :

amlxl + ...... + a mm=x iln-8 has a solution if and only if the column vector lies

in the space generated by the vectors

(..)- (i:-) 10

This means that there is a solution if and only if the right column rank of the matrix 51.. .%n is the same as the

iia ml’ . . amn4 right column rank of the augmented matrix 51. . * %A ......

i!a ml* * ’ arnnb, i since the vector space generated by the original must be the same as the vector space generated by the augmented matrix and in either case the dimension is the same as the rank of the matrix by Theorem 2. By Theorem 4, this means that the row tanks are equal. Con- versely, if the row rank of the augmented matrix is the same as the row rank of the original matrix, the column ranks Will be the same and the equations Will have a solution. If the equations (2) have a solution, then any relation among the rows of the original matrix subsists among the rows of the augmented matrix. For equations (2) this merely means that like combinations of equals are equal. Conversely, if each relation which subsists be- tween the rows of the original matrix also subsists between the rows of the augmented matrix, then the row rank of the augmented matrix is the same as the row rank of the original matrix. In terms of the equations this means that there Will exist a solution if and only if the equations are consistent, Le., if and only if any dependence between the left hand sides of the equations also holds between the right sides. 11

THEOREM 5. If in equations (2) m = n, there exists a unique solution if and only if the corresponding homogeneous equations arrxr + arzxz + . . . + alnxn = 0

anlxl + an2x2 + . . . + annxn = 0 have only the trivial solution. If they have only the trivial solution, then the column vectors are independent. It follows that the original n equations in n unknowns Will have a unique solution if they have any solution, since the differ- ence, term by term, of two distinct solutions would be a non-trivial solution of the homogeneous equations. A solution would exist since the n independent column vectors form a generating system for the n-dimensional space of column vectors. Conversely, let us suppose our equations have one and only one solution. In this case, the homogeneous equations added term by term to a solution of the original equations would yield a new solu- tion to the original equations. Hence, the homogeneous equations have only the trivial solution.

F. Qeterminants. l) The theory of determinants that we shall develop in this chapter is not needed in Galois theory. The reader may, therefore, omit this

section if he SO desires. We assume our field to be c o m m ut a t i v e and consider the square matrix 1) Of the preceding theory only Theorem 1, for homogeneous equations and the notion of linear dependence are assumed known. 12

allalz* . . . %n

a21a22~**.% (1) ......

an1 an2. . . . a nn of n rows and n columns. We( shall define a Jcertain function of this matrix whose value is an element of our field. The function Will be called the determinant and Will be denoted by

%1%2. . . .%n

%!1a22--.‘a2n

......

n,?l,* *. ‘?ln orby D(A,,A,,... An) if we wish to consider it as a function of the column vectors A,, A,, . . . A,, of (1). If we keep a11 the columns but A, constant and consider the determinant as a function of A,, then we

Write DJ Ak) and sometimes even only D.

Definition. A function of the column vectors is a determinant if it satisfies the following three axioms:

1. Viewed as a function of any column A, it is linear and homogeneous, i.e..,

(3) &(A, + 4) = Dk(Ak) + &(A;)

(4) D,(cA,) = c-D,(A, >

2. Its value is = 01) if the adjacent columns A, and Ak+l are equal.

3. Its value is = 1 if a11 A, are the unit vectors U, where

1) Henceforth, 0 Will denote the zero element of a field. (5) “, f);“*=(I) . . . . . ;-iii

The question as to whether determinants exist Will be left open for the present. But we derive consequences from the axioms:

a) If we put c = 0 in (4) we get: a determinant is 0 if one of the columns is 0.

or a determinant remains unchanged b) Dk(Ak) = &(A, + CA,,,)- if we add a multiple of one column to an adjacent column. Indeed

%(A, + CA,,,)- = Dk(Ak) + cD,(A,+,) = Dk(Ak) because of axiom 2.

c) Consider the two columns A, and Ak+i. We may replace them by

A, and Ak+i + Ak; subtracting the second from the first we may replace

them by - Ak+i and Ak+i + A,,. adding the first to the second we now have - Ak+r and A,,. finally, we factor out -1. We conclude: a determi-

nant changes sign if we interchange two adjacent columns.

d) A determinant vanishes if any two of its columns are equal.

Indeed, we may bring the two columns side by side after an interchange

of adjacent columns and then use axiom 2. In the same way as in b)

and c) we may now prove the more general rules:

e) Adding a multiple of one column to another does not change

the value of the determinant.

f) Interchanging any two columns changes the sign of D. 14

g> Let(v,,v,,... vn) be a permutation of the subscripts (1,2,, . . . n). If we rearrange the columns in D( At,,i, AV2, . . . , A,, ) n until they are back in the natural order, we see that WAvl,Av >...y A, ) = +D(A,,A, ,...> An). 2 Here 2 is a definite sign that do:s not depend on the special values of the A,. If we substitute U, for A, we see that WJl,JJv2,..., U, ) = i 1 and that the sign depends only on the 1 permutation of the tnit vectors. Now we replace each vector A, by the following linear combina- tionAkofAr,A2,...,A,: (6) A; = brkA, + b,,A, + . . . + b,,A,. In computing D(Ai ,A;, . . . , AA) we first apply axiom 1 on A; breaking up the determinant into a sum; then in each term we do the

same with A; and SO on. We get (7) D(A;,A;,...,A;)= 2 D(b, ,A, ,bv22Av >. . . Jj, ,AI, > v1 ,v2*. ,v, 1 1 2 n n = c b, ;b, 2.. . ..b,nD(Ar, ,A,, . . . ,A, ) 1 2 n 1 2 I-4 VI,V2, * *. ,vn where each vi runs independently from 1 to n. Should two of the indices vi be equal, then D( Avl, A, , . . . , AV,) = 0; we need therefore keep 2 only those terras in which ( vi, v2, . . . , vn) is a permutation of (1,2,..., n). This gives (8) D(A;,A;,...,A;) = D(A,,A2 ,...> A,). 2 + bv1,.bv2, . . . . .b, n ( vl, * * * 9Vn) n where(v1,v2,..., v,) runs through a11 the permutations of (1,2,..., n) and where L stands for the sign associated with that permutation. It is important to remark that we would have arrived at the same formula (8) if our function D satisfied only the first two 1.5 of our axioms.

Many conclusions may be derived from (8).

W’e first assume axiom 3 and specialize the 4, to the unit vec- tors Uk of (5). This makes 4 = B, where B, is the column vector of the matrix of the bik. (8) yields now:

(9) WBl,B2,...,B,,)=(V 2 + bVll+V2î-bv n n 1 >UZ’...’ vn > - giving us an explicit formula for determinants and showing that they are uniquely determined by our axioms provided they exist at all.

W’ith expression (9) we retum to formula (8) and get

(10) D(A;,A; ,...> A;) = D(A,,A, ,...> A,)D(B,,B, ,...> Bn).

This is the so-called multiplication theorem for determinants. At the left of (10) we have the determinant of an n-rowed matrix whose ele- ments cik are given by n (11) Cik = x ?vbw v=1 cik is obtained by multiplying the elements of the i - th row of

WA,,A,,..., AJbythoseofthek-thcolumnofD(B,,B,,...,B,.,)

and adding.

Let us now replace D in (8) by a function F(A,, . . . , A,) that

satisfies only the first two axioms. Comparing with (9) we find

F(A;,A; ,..., A;)=F(A, >..., AJD(BI,B2 ,...> B,).

Specializing A, to the unit vectors U, leads to

(12) F(B,,B,, . . . ,B,) = c.D(B,,B,, . . . ,B,)

with c = F(U,,U,,. . . ,U”). 16

Next we specialize (10) in the following way: If i is a certain subscript from 1 to n-l we put A, = U, for k f i, i+ 1 Ai = Ui + Ui+r , Ai+, = 0. Then D( A,, A,, + . . , A, ) = 0 since one col- umn is Q, Thus, D(Ai ,A;, , . . , An) = 0; but this determinant differs from that of the elements bj, only in the respect that the i+l-st row has been made equal to the i-tb. We therefore see:

A determinant vanishes if two adjacent rows are equal. I&ch term in (9) is a product where precisely one factor cornes from a given row, say, the i-th. This shows that the determinant is linear and homogeneous if çonsidered as function of this row. If, finally, we Select for eaeh raw the corresponding unit vector, the de- terminant is = 1 since the matrix is the same as that in which the col- umns are unit vectors. This shows that a determinant satisfies our three axioms if we consider it as function of the row vectors. In view of the uniqueness it follows: A determinant remains unchanged if we transpose the row vec- tors into column vectors, that is, if we rotate the matrix about its main diagonal. A determinant vanishes if any two rows are equal. It changes sign if we interchange any two rows. It remains unchanged if we add a multiple of one row to another.

We shall now prove the existence of determinants. For a 1-rowed

matrix a 1 1 the element ai 1 itself is the determinant. Let us assume the existence of (n - 1) - rowed determinants. If we consider the n-rowed matrix (1) we may associate with it certain (n - 1) - rowed determinants in the following way: Let ai, be a particular element in (1). We 17 cancel the i-th row and k-th column in (1) and take the determinant of the remaining (n - 1) - rowed matrix. This determinant multiplied by (-l)i+k Will be called the cofactor of a ik and be denoted by Ai,. The distribution of the sign (- 1) i+k follows the chessboard pattern, namely,

...... Let i be any number from 1 to n. We consider the following function D of the matrix (1):

(13) D = ailAi, + ai2Ai, + . . + ainAi,. [t is the sum of the products of the i-th Tow and their cofactors. Consider this D in its dependence on a given column, say, A,. For v f k, Au, depends linearly on A, and ai, does not depend on it; for v =: k, Ai, does not depend on A, but aik is one element of this column. Thus, axiom 1 is satisfied. Assume next that two adjacent columns A, and Ak+l are equal. For v f k, k + 1 we have then two equal columns in Ai, SO that A,, = 0. The determinants used in the computation of Ai k and Ai k+l are the same but the signs are opposite hence, Ai k = -Ai k+l whereas ai k = a, k+l’ Thus D = 0 and axiom 2 holds. For the special case A, = U,( v = 1,2,. . , n) we have aiV = 0 for v f i while a,, = 1, Aii = 1. Hence, D = 1 and this is axiom 3. This proves both the existence of an n-rowed 18 determinant as well as the truth of formula (13), the so-called develop- ment of a determinant according to its i-th row. (13) may be generalized as follows: In our determinant replace the i-th row by the j-th row and develop according to this new row. For i f j that determinant is 0 and for i = j it is D: D for j = i (14) + ajzAi2 t . . . + ainAi,, = ajl *il 0 forj f i

If we interchange the rows and the columns we get the following formula: D for h = k (15) a,,* Ik t aZr,A,, + . . + a,hAnk = 0 for h f k Now let A represent an n-rowed and B an m-rowed square matrix. By ( A 1, ( B \ we mean their determinants. Let C be a matrix of n rows and m columns and form the square matrix of n + m rows

where 0 stands for a zero matrix with m rows and n columns. If we con- sider the determinant of the matrix (16) as a function ofthecolumns of A only, it satisfies obviously the first two of our axioms. Because of (12) its value is c . 1 A 1 where c is the determinant of (16) after substituting unit vectors for the columns of A. This c still depends on B and con- sidered as function of the rows of B satisfies the first two axioms. Therefore the determinant of (16) is d. 1 A 1 . 1 B 1 where d is the special case of the determinant of (16) with unit vectors for the columns of A as well as of B. Subtracting multiples of the columns of A from C we cari replace C by 0. This shows d = 1 and hence the formula 19

(17)

In a similar fashion we could have shown (18) A 0 = \A\. JBI. I C BI The formulas (17), (18) are special cases of a general theorem by Lagrange that cari be derived from them. We refer the reader to any textbook on determinants since in most applications (17) and (18) are sufficient. We now investigate what it means for a matrix if its determinant is zero. We cari easily establish the following facts: a) If A,, A,, . , . , An are linearly dependent, then DCA,, A,, . . . t A,) = 0. Indeed one of the vectors, say A,, is then a linear combination of the other columns; subtracting this linear com- bination from the column A, reduces it to 0 and SO D = 0. b) If any vector B cari be expressed as linear combination of A,, A,, . . . >A, then D(A,,A,,. . ., A,,) # 0. Returning to (6) and (10) we may Select the values for bi, in such a fashion that every A! ,= I.Ji. For this choice the left side in (10) is 1 and hence DCA,,&..., A,) on the right side f 0. c) Let A,, A,, . . . , A,, be linearly independent and B any other vector. If we go back to the components in the equation Aix, + A,x, + . . . + A,.,x,,+ By = 0 we obtain n linear homogeneous equations in the n + 1 unknowns x i, x 2,. . . , xn, y. Consequently, there is a non-trivial solution. y must be f 0 or else the ApAz,...,& would be linearly dependent. But then we cari compute B out of this equation as a linear combination of A,, A,, . . . , An. 20

Combining these results we obtain: A determinant vanishes if and only if the column vectors (or the row vectors) are linearly dependent. Another way of expressing this result is: The set of n linear homogeneous equations

(i = 1,2,...,n) ail 3 + ai2x2 + . . . + ainx n = 0 in n unknowns has a non-trivial solution if and only if the determinant of the coefficients is zero. Another result that cari be deduced is: If A1,A2>..., A,, are given, then their linear combinations cari represent any other vector B if and only if D (A *, A,, . . . , An) f 0.

Or: The set of linear equations

(19) aiixI + ai2x2 + . . . + ainxn = bi (i = 1,2,...,n)

has a solution for arbitrary values of the bi if and only if the determi- nant ‘of aik is f 0. In that case the solution is unique. We finally express the solution of (19) by means of determinants

if the determinant D of the aik is f 0. We multiply for a given k the i-th equation with Ai, and add the equations. (15) gives (20) D. xk = A,,b, + A,,bz + + Ankb, (k = 1,2,...,n) and this gives xk. The right side in (12) may also be written as the determinant obtained from D by replacing the k-th column by b,, b,, . . , b”. The rule thus obtained is known as Cramer’s rule. 21

II FIELD THEORY

A. -Extension Fields. If E is a field and F a subset of E which, under the operations of addition and multiplication in E, itself forms a field, that is, if F is a subfield of E, then we shall cal1 E an extension of F. The relation of being an extension of F Will be briefly designated by F C E. If a, P, y, . . . are elements of E, then by F(a, B, y, . . . ) we shall mean the set of elements in E which cari be expressed as quotients of poly- nomials in a, p, y, . . with coefficients in F. It is clear that

F(a,/3,y,. . . ) is a field and is the smallest extension of F which con- tains the elements a, p, y,. . We shall cal1 F(a, 6, y,. . . ) the field obtained after the adjunction of the elements a, @, y, . . . to F, or the field generated out of F by the elements a, B, y, . . . . In the sequel a11 fields Will be assumed commutative. . . ~. If F C E, then ignoring the operation of multiplication defined between the elements of E, we may consider E as a vector space over F. By the degree of E over F, written (E/F), we shall mean the dimen- sion of the vector space E over F. If (E/F) is finite, E Will be called a finite extension. THEOREM 6. If F, B, E are three fields such that

F C ES C E, then WF) = (B/F) (E/B) . Let A1,A2,..., A, be elements of E which are linearly independent with respect to B and let C 1, C,, . . . , C s be elements 22

of B which are independent with respect to F. Then the products Ci Ai where i = 1,2, . . . , s and j = 1,2, . . . , r are elements of E which are independent with respect to F. For if 2 arj C,A, = 0, then Lj C( iajj Ci ) Aj is a linear combination of the A, with coefficients in B j and because the Aj were independent with respect to B we have pij Ci = 0 for each j. The independence of the Ci with respect to F then requires that each aij = 0. Since there are r . s elements C,A, we have shown that for each r 5 (E/B) and s 5 (B/F) the degree ( E/F )

-> r . s. Therefore, ( E/F) >- (B/F) ( E/B). If one of the latter numbers is infinite, the theorem follows. If both (E/B) and (B/F) are finite, say r and s respectively, we may suppose that the Aj and the Ci are generating systems of E and B respectively, and we show that the set of products Ci Aj is a generating system of E over F. Each A E E cari be expressed linearly in terms of the Aj with coefficients in B. Thus, A = CBj Aj . Moreover, each Bj being an element of B cari be ex- pressed linearly with coefficients in F in terms of the Ci, i.e., Bj = Caij Ci, j = 1,2, . . . ,r. Thus, A = Xaij CiAj and the Cil form an independent generating system of E over F.

--Corollary. If F C Fi C F, C . . . C F,, then (Fn/F) =y (F,/F).(F,/F, > . . . (F,,/F,,i).

B. Polvnomials. An expression of the form aOxn + a ix”-i+ . . . + an is called a

-~-polynomial in F of degree n if the coefficients 23 a 01. . . > a,., are elements of the field F and ao f 0. Multiplication and addition of polynomials are performed in the usual way ‘). ~4 polynomial in F is called reducible in F if it is equal to the product of two polynomials in F each of degree at least one. Polyno- mials which are not reducible in F are called irreducible in F. If f (x ) = g(x) . h (x ) is a relation which holds between the polynomials f (x ), g (x ), h (x ) in a field F, then we shall say that g (x ) divides-- f (x ) in F, or that g ( x ) is a factor of f ( x ). It is readily seen that the degree of f(x) is equal to the sum of the degrees of g (x ) and h (x ), SO that if neither g ( x ) nor h ( x ) is a constant then each has a degree less than f(x). It follows from this that by a finite number of factorizations a polynomial cari always be expressed as a product of irreducible polynomials in a field F. For any two polynomials f (x ) and g (x ) the division algorithm holds, i.e., f(x) = q(x).g(x) + r(x) where q(x) and r(x) are unique polynomials in F and the degree of r (x ) is less than that of g(x). ‘This may be shown by the same argument as the reader met in elementary algebra in the case of the field of real or complex numbers. We also see that r(x) is the uniquely determined polynomial of a de- gree less than that of g (x ) such that f(x) - r (x ) is divisible by g (x ). We shall cal1 r (x ) the remainder of f (x ).

1) If we speak of the set of a11 polynomials of degree lower than II, we shall agree to include the polynomial 0 in this set, though it has no degree in the proper sense. 24

Also, in the usual way, it may be shown that if a is a root of the polynomial f (x ) in F than x - u is a factor of f (x ), and as a con- sequence of this that a polynomial in a field cannot have more roots in the field than its degree. --Lemma. If f(x) is an irreducible polynomial of degree n in F, then-- there do not exist two polynomials each of degree less than n in --F whose product is divisible by f(x). Let us suppose to the contrary that g(x) and h(x) are poly- nomials of degree less than n whose product is divisible by f(x). Among a11 polynomials occurring in such pairs we may suppose g(x) has the smallest degree. Then since f(x) is a factor of g(x) . h (x ) there is a polynomial k(x) such that k(x).f(x) = g(x).h(x) By the division algorithm, f(x) = q(x).g(x) + r(x) where the degree of r (x ) is less than that of g(x) and r (x ) f 0 since f(x) was assumed irreducible. Multiplying f(x) = q(x).g(x) + r(x) by h (x ) and transposing, we have r(x),h(x) = f(x).h(x)-q(x).g(x).h(x)=f(x).h(x)-q(x).k(x).f(x) from which it follows that r(x) . h (x ) is divisible by f (x ). Since r (x ) has a smaller degree than g(x), this last is in contradiction to the choice of g (x ), from which the lemma follows. As we saw, many of the theorems of elementary algebra hold in any field F. However, the so-called Fundamental Theorem of Algebra, at least in its customary form, does not hold. It Will be replaced by a theorem due to Kronecker 25 which guarantees for a given polynomial in F the existence of an ex- tension field in which the polynomial has a root. We shall also show that, in a given field, a polynomial cari net only be factored into irre- ducible factors, but that this factorization is unique up to a constant factor. The uniqueness depends on the theorem of Kronecker.

C. Algebraic Elements.

Let F be a field and E an extension field of F. If a is an ele-

ment of E we may ask whether there are polynomials with coefficients in F which have a as root. a ia çalled algebraic.- with respect to F if tkere are such polynomials. New let a be algebraic and Select among ail polynomials in F which have a as root one, f(x), of lowest degree.

We may assume that the highest coefficient of f(x) La 1. We con- tend that this f(x) ia uniquely determined, that it ts trreducible and that each polynomial in F w#r the root o is divisible by f (x ). If, in-

deed, g ix ) !w a palynomial in F with g(a) = 0, we may divide

g(x) == f(x)q(x) t r(x) where r(x) bas a degree smaller than that of f(x). Substituting x = a we get r(o) = Q: Dow r(x) has to he identically 0 since otherwise r (x > would havg the root a apd be of lower degree thap f (x ): SO g ( x ) ia divisible by f (x )! Thia also shows the uniqueness of f (x ). If f (x ) were not irreducible, one of the factors

wopld have to vanish for x = a contradicting again the choice of f ( y ).

We consider now the subset E0 of the following elements 8 of E: 26

8 = g(a) = CO + cla + c2a2 + . . . + CnTlanel where g(x) is a polynomial in F of degree less than n (n being the de- gree of f(x)). This set l$, is closed under addition and multiplication. The latter may be verified as follows: If g (x ) and h (x ) are two polynomials of degree less than n we put g(x)h(x) = q(x)f(x) + r(x) and hence g(a)h(a) = r(a). Finally we see that the constants cO, c 1, . . , cr,i are uniquely deter- mined by the element 8. Indeed two expressions for the same 0 would lead after subtracting to an equation for a of lower degree than n. We remark that the interna1 structure of the set EO does not de- pend on the nature of a but only on the irreducible f (x ). The knowledge of this polynomial enables us to perform the operations of addition and multiplication in our set EO. We shall see very soon that E, is a field; in fact, EO is nothing but the field F(a). As soon as this is shown we have at once the degree, ( F (a) /F), determined as n, since the space F(a) is generated by the linearly independent 1, a, a2, . . . , an-l. We shall now try to imitate the set EO without having an exten- sion field E and an element a at our disposal. We shall assume only an irreducible polynomial

f(x) = x” + an-lxn-i + . . . + aO as given. We Select a symbol 6 and let E, be the set of a11 forma1 polynomials g(5‘) = CO + c,c + . . + cnJy-l of a degree lower than n. This set forms a group under addition. We now introduce besides the ordinary multiplication 27 a new kind of multiplication of two elements g (5) and h (4) of E i denoted by g ([) x h (5). It is defined as the remainder r (6) of the ordinary product g (6) h(c) un d er d ivision by f (4‘ ). We first remark that any product of m terms gi( c), gz( t), . . . , g,( 0 is again the re- mainder of the ordinary product g i( 5) g,( 5). . . g,( 5). This is true by definition for m = 2 and follows for every m by induction if we just prove the easy lemma: The remainder of the product of two remainders (of two polynomials) is the remainder of the product of these two polynomials. This fact shows that our new product is associative and commutative and also that the new product g i( 4) x g,( 4) x . . . x g I[) Will coincide with the old product g i( 5) g,( 6). . . g,( 6) if the latter does not exceed n in degree. The distributive law for our multiplication is readily verified. The set E i contains our field F and our multiplication in E, has for F the meaning of the old multiplication. One of the polynomials of E, is ç:. Multiplying it i-times with itself, clearly Will just lead to ti as long, as i < n. For i = n this is not any more the case since it leads to the remainder of the polynomial 5”. This remainder is

5” - f(t) = - a,-&“-‘- anJn-*- . . . - a,. We now give up our old multiplication altogether and keep only the new one; we also change notation, using the point (or juxtaposition) as symbol for the new multiplication. Computing in this sense

c, + Cl[ + c*p + . . . + c,-lp-l Will readily lead to this element, since a11 the degrees 28 involved are below n. But 5” = - anyl[n-l- a,-2[n-2- . . . - a0. Transposing we see that f(ç) = 0. We thus have constructed a set E, and an addition and multipli- cation in E r that already satisfies most of the field axioms. E r contains F as subfield and 5‘ satisfies the equation f (5) = 0. We next have to show: If g ( 6) $ 0 and h ( .$) are given elements of E r, there is an element X(l$> = x, + x1( + . . . + X,J--1 in E, such that g(Ç) *X(t) = h(t).

TO prove it we consider the coefficients xi of X (6) as unknowns and compute nevertheless the product on the left side, always reducing higher powers of [ to lower ones. The result is an expression L, + LJ + . . + L,-, (““where each Li is a linear combination of of the xi with coefficients in F. This expression is to be equal to h(t); this leads to the n equations with n unknowns: L, = b,, L, = b,, . . . > L,-, = b,-, where the bi are the coefficients of h(E). This system Will be soluble if the corresponding homogeneous equations L, = 0, L, = 0, * . . > L,-r = 0 bave only the trivial solution. The homogeneous problem would occur if we should ask for the set of elements X(Q) satisfying g (5) . X ( 6) = 0. Going back for a moment to the old multiplication this would mean that the ordinary product g( 6) X (6) has the remainder 0, and is 29 therefore divisible by f(t). According to the lemma, page 24, this is only possible for X (6) = 0.

Therefore E, is a field.

Assume now that we have also our old extension E with a root a of f(x), leading to the set E,. We see that E, has in a certain sense the same structure as E 1 if we map the element g (6) of E 1 onto the element g(a) of EO. This mapping Will have the property that the image of a sum of elements is the sum of the images, and the image of a product is the product of the images.

Let us therefore define: A mapping u of one field onto another which is one to one in both directions such that o(a+~) = o(a) + CT(~) and O(U.@) = o(a). o(p) is called an

~-isomorphism. If the fields in question are not distinct - i.e., are both the same field - the isomorphism is called an automorphism. Two fields for which there exists an isomorphism mapping one on another are called isomorphic. If not every element of the image field is the image under o of an element in the first field, then 0 is called an isomorphism of the first field into the second. Under each isomorphism it is clear that o(O) = 0 and o( 1) = 1. We see that E, is also a field and that it is isomorphic to E,.

We now mention a few theorems that follow from our discussion:

THEOREM 7. (Kronecker). If f (x ) is a polynomial in a field F, there exists an extension E of F in which f(x) has a root. 30

Proof: Construct an extension field in which an irreducible factor of f ( x ) has a root.

THEOREM 8. Let o be an isomorphism mapping a field F on a

~~field F’ Let f (x ) be an irreducible polynomial in F and f ’ (x ) the cor-

-~responding polynomial in F ’ . If E = F (B) and E ’ = F ’ (@‘) are exten- ~~sions of F and F’ , respectively, where f(p) = 0 in E and f ’ ( p ‘) = 0 in E’ , then o’ cari be extended to an isomorphism between E and E ’ . Proof: E and E’ are both isomorphic to EO.

D. Splitting Fields. If F, B and E are three fields such that F C B C E, then we shall refer to B as an intermediate field. If E is an extension of a field F in which a polynomial p(x) in F cari be factored into linear factors, and if p(x) cari not be SO factored in any intermediate field, then we cal1 E a splitting field for p(x). Thus, if E is a splitting field of p(x), the roots of p(x) generate E. A splitting field is of finite degree since it is constructed by a finite number of adjunctions of algebraic elements, each defining an extension field of finite degree. Because of the corollary on page 22, the total degree is finite.

-~~THEOREM 9. If p(x) is a polynomial in a field F, there exists

~~a splitting field E of p(x). We factor p (x ) in F into irreducible factors f,(x) . . . f*(x) = p(x). If each of these is of the first degree then F itself is the required splitting field. Suppose then that fi(x) is of degree higher than the first. By 31

Theorem 7 there is an extension Fr of F in which f r( x ) has a root. Factor each of the factors f r( x), . . . , fr( x ) into irreducible factors in Fr and proceed as before. We finally arrive at a field in which p (x) cari be split into linear factors. The field generated out of F by the roots of p(x) is the required splitting field.

The following theorem asserts that up to isomorphisms, the splitting field of a polynomial is unique.

THEOREM 10. Let (T be an isomorphism mapping the field F on

~~the field F’ , Let p (x ) be a polynomial in F and p ’ (x ) the polynomial

--~in F ’ with coefficients corresponding to those of p (x ) under 0. Finally,

~-let E be a splitting field of p(x) and E’ a splitting field of p’ (x). Under~~ these conditions the isomorphism o cari be extended to an isomorphism between E and E’ .

If f(x) is an irreducible factor of p(x) in F, then E contains a root of f( x ). For let p (x )=(x-a J (x-a, ) . . (x-a .) be the splitting of p(x) in E. Then (x-ar)(x-a,). . .(x-as) = f(x) g(x). We consider f(x) as a polynomial in E and construct the extension field B = E(a) inwhichf(a) = 0. Then(a-aI).(a-a2)...:(a-as) = f(a).g(a) = 0 and a-ai being elements of the field B cari have a product equal to 0 only if f’or one of the factors, say the first, we have a-a1 = 0. Thus, a = al, and a1 is aroot of f(x).

Now in case a11 roots of p(x) are in F, then E = F and p(x) cari be split in F. This factored form has an image in F’ which is a splitting, of p’ (x), since the isomorphism o preserves a11 operations of addition and multiplication in the process of multiplying out the factors of p(x) and collecting to get the original form. Since p ’ (x) cari be split in F’ , we must have F ’ = E ’ . In this case, o itself is the required extension and the theorem is proved if a11 roots of p(x) are in F. We proceed by complete induction. Let us suppose the theorem proved for a11 cases in which the number of roots of p(x) outside of F is less than n > 1, and suppose that p (x ) is a polynomial having n roots outside of F. We factor p (x ) into irreducible factors in F; p(x) = f,(x) fJx) . . f,(x). Not a11 of these factors cari be of degree 1, since otherwise p (x ) would split in F, contrary to assump- tion. Hence, we may suppose the degree of f 1( x) to be r > 1. Let f’,(x).f\(x) . . . f;(x) = p’(x) be the factorization of p’(x) into the polynomials corrrespondng to f 1( x ) , . . . , fm( x ) under O. fi (x ) is irreducible in F ’ , for a factorization of fi (x) in F ’ would induce 1) under 0-l a factorization of f,(x), which was however taken to be irreducible. By Theorem 8, the isomorphism o cari be extended to an isomor- phism ol, between the fields F(a) and F ’ (a’ ). Since F C F(a), p(x) is a polynomial in F(U) and E is a splitting field for p(x) in F(a). Similarly for p ’ (x). There are now less than n roots of p (x ) outside the new ground field F (a). Hence by our inductive assumption o1 cari be extended from an isomorphism between F(a) and F ’ (a ’ ) to an isomorphism o2 between E and E ’ . Since u, is an extension of (T, and o2 an extension of o,, we conclude u2 is an extension of u and the theorem follows.

1) See page 38 for the definition of (2-l. 33

-~Corollary. If p(x) is a polynomial in a field F, then any two --splitting fields for p (x ) are isomorphic. This follows from Theorem 10 if we take F = F ’ and o to be the identity mapping, i.e., o(x) = x. As a consequence of this corollary we see that we are justified in using the expression ‘?he splitting field of p(x)” since any two differ only by an isomorphism. Thus, if p (x ) has repeated roots in one splitting field, SO also in any other splitting field it Will have repeated roots. The statement “p(x) has repeated roots” Will be significant without reference to a particular splitting field.

E. Unique--- Decomposition of Polynomials into Irreducible Factors. --THEOREM 11. If p(x) is a polynomial in a field F, and if ~-p(x) = pi(x).p,(x)... . .p,(x) = qi(x).q*(x)... . . qs(x) are two factorizations of p(x) into irreducible polynomials each of degree at

~-least one, then r = s and after a suitable change in the order in which the q’s are written, p,(x) = ciqi(x), i = 1,2,. . ,r, and ci 6 F. ~- Let F(a) be an extension of F in which p i(a) = 0. We may suppose the leading coefficients of the pi( x ) and the qi( x ) to be 1, for, by factoring out a11 leading coefficients and combining, the constant multiplier on each side of the equation must be the leading coefficient of p (x ) and hence cari be divided out of both sides of the equation. Since 0 = ~~(a).~~(a). . . . .~,(a) = p(a) = si(a).. . . .4,(a) and since a product of elements of F(a) cari be 0 only if one of these is 0, it follows that one of the qi( a), say qi( a), is 0. This gives (see page 25)p,(x) = si(x). Thus~,(x).~,(x).....~,(x) = Pdx).q,(x). . . .q,(x) or 34 pi(x).[p,(x).. . . .p,(x) - q*(x).. . . .qs(x)] = 0. Since the product of two polynomials is 0 only if one of the two is the 0 polynomial, it follows that the polynomial within the brackets is 0 SO that p,(x) . . . .p,(x) = q*(x). . .: q.(x). If we repeat the above argument r times we obtain p,(x) = si(x), i = 1,2,. . , r. Since the remaining q’s must have a product 1, it follows that r = s.

F. Group-~- Characters.-- If G is a multiplicative group, F a field and o a homomorphism mapping G into F, then o is called a character of G in F. By homomor- phism is meant a mapping u such that for a, fi any two elements of G, o(a).a(B) = a(a.@)ando(a) f Oforanya.

(If o(a) = 0 for one element a, then o(x) = 0 for each x t G, since o( ay) = o(a). o(y) = 0 and ay takes a11 values in G when y assumes a11 values in G).

The characters or, 02,. . . , onare called dependent if there exist elements a r, a,, . . . , a,, not a11 zero in F such that a,o,(x) + a202(x) + . . . + anon = 0 for each x t G. Such a de- pendence relation is called non-trivial. If the characters are not dependent they are called independent.

THEOREM 12. If G is a group and or, u2,. . . , on are n mutu- ally distinct characters of G in a field F, then oi, 02,. . . , on are independent.

One character cannot be dependent, since a rcr( x) = 0 implies a1 = 0 due to the assumption that or(x) f 0. Suppose n > 1. 35

We make the inductive assumption that no set of less than n distinct characters is dependent. Suppose now that aru, i a,o,(x> + . . + angn( x) = 0 is a non-trivial dependence between the u’s. None of the elements ai is zero, else we should have a dependence between less than n characters contrary to our induc- tive assumption. Since or and un are distinct, there exists an element a in G; such that or (a) f o”(a). Multiply the relation between the u’s byn a-rWe obtain a relation (*) bru,(x) + . . . + b,.r on-r(x) + o,(x) = 0, bi = air ai f 0. Replace in this relation x by ax. We have b,o,(a)ol(x) + . . + b,-, un., (a>un.,(x> + un(a (x> = 0, or a, (a j’b,u,(a)u, (x) + . + U,(X) = 0. Subtracting the latter from (*) we have

(**> [b, - un (a)-‘blul (a>la,(x) t- . + cn.lun.l (x) = 0. The c’oefficient of ur (x ) in this relation is not 0, otherwise we should have b, = u, (a)-‘b,al (a), SO that

q, (a)b, = blo,(a) = u,(a)b, and since b, f 0, we get a,( a) = ur (a) contrary to the choice of a. Thus, (* * ) is a non-trivial dependence between u r, g2, . ,v”- 1 which is contrary to our inductive assumption.

Corollary. If E and E’ are two fields, and q , u2, . . , un are n mutually distinct isomorphisms mapping E into E ’ , then u, , . , u, are independent. (Where “independent” again means there exists no non-trivial dependence a ru r (x ) + . + anun (x ) = 0 which holds for every x 6 E). This follows from Theorem 12, since E without the 0 is a group 36 and the u’s defined in this group are mutually distinct characters.

If oi > a2 > . . . , u, are isomorphisms of a field E into a field E’ , then each element a of E such that o*(a) = o,(a) = . . . = on(a) is called a fixed point of E under oi , 02, . . . , o,., . This name is chosen because in the case where the u’s are automorphisms and ui is the identity, i.e., u1 (x) = x, we have ui (x) = x for a fixed point.

Lemma. The set of fixed points of E is a subfield of E. We shall cal1 this subfield the fixed field.

For if a and b are fixed points, then ui(a -1 b) = u,(a) + u,(b) = uj (a) + oj (b) = uj (a + b) and uj(a.b) = ui(a).ui(b) = uj (a).uj(b) = uj (a.b).

Finally from u,(a) = aj (a) we have (uj(a))-’ = (u,(a))-’

= ~,(a-‘) = uj (a-‘).

Thus, the sum and product of two fixed points is a fixed point, and the inverse of a fixed point is a fixed point. Clearly, the negative of a fixed point is a fixed point.

THEOREM 13. If cri,. . . >un are n mutually distinct isomorphisms of a field E into a field E’ , and if F is the fixed field of E, then

(E/F:) L > n. Suppose to the contrary that (E/F) = r < n. We shall show that we are led to a contradiction. Let w i, o 2, . . . , o, be a generating sys- tem of E over F. In the homogeneous linear equations 37

ul(ol)xl + u*(w1)x2 + . . . + u,(wl)x, = 0 OI(OZ)XI + u2(o*)x2 + . . . + u,(o*)xn = 0

UI(W,)XI + u*(or)x2 + . . . + un(o,)xn = 0 there are more unknowns than equations SO that there exists a non- trivial solution which, we may suppose, x r, x 2, . . . ,x, denotes. For any element a in E we cari find ar, a2,. . . , a, in F such that a = aIci>, f . . . + a,o,. We multiply the first equation by o 1 ( a 1 ), the second by o1 (a ,J, and SO on. Using that ai 6 F, hence that ol(ar) = oj (ai)and also that oj(ai) oj(ai) = oj(aiwi), we obtain

ol(a,wI)x, + . . . + on(a,o,)xn = 0

oI(ar~,)xl + . . . + on(arWr)x n = 0. Adding these last equations and using oi(a,o,) + oi(a2W2) + . . . + oi(aro,) = oi(a,o, + . . . + aru,) = ai(Q) we obtain o,(a)x, + o,(a>x, + . . . + un(a)xn = 0. This, however, is a non-trivial dependence relation between or, 02, . . . , on which cannot exist according to the corollary of Theorem 12.

-Ciorollary. If or, u2, . . . , cr,, are automorphisms of the field E, and F is the fixed field, then (E/F) >- n. If F is a subfield of the field E, and 0 an automorphism of E, we shall say that u leaves F fixed if for each element a of F, o(a) = a. 38

If o and r are two automorphisms of E, then the mapping o( r( x)) written briefly ur is an automorphism, as the reader may readily verify. [E.g., u~(x.Y) =o(r(~.~)) = o(r(~>.~(y)> = a(~(x))~o(s(y))l. We shall cal1 UT the product of o and r. If o is an automorphism

(o(x) = y), then we shall cal1 0-l the mapping of y into x, i.e.,o-‘(y) = x the inverse of o. The reader may readily verify that o-r is an automor- phism. The automorphism 1 (x ) = x shall be called the

--unit automorphism.-- -~Lemma. If E is an extension field of F, the set G of automorphisms which leave F fixed is a group. The product of two automorphisms which leave F fixed clearly leaves F fixed. Also, the inverse of any automorphism in G is in G. The reader Will observe that G, the set of automorphisms which leave F fixed, does not necessarily have F as its fixed field. It may be that certain elements in E which do not belong to F are left fixed by every automorphism which leaves F fixed. Thus, the fixed field of G may be larger than F.

G. Applications and Examples to Theorem 13.

Theorem 13 is very powerful as the following examples show:

1) Let k be a field and consider the field E = k (x ) of a11 rational functions of the variable x. If we map each of the functions f (x ) of E onto f(L) we obviously obtain an automorphism of E. Let us X consider the following six automorphisms where f (x ) is mapped onto f (x ) (identity), f ( l-x), f ($), f (l-i), f (&) and f (5) and cal1 F the 39 fixed point field. F consists of a11 rational functions satisfying

(1) f(x) = f(l-x) = f(i) = f(l-$) = f(A) = f($$. It suffices to check the first two equalities, the others being conse- quences. The function

(2) 1 = I(x) = (x2 - x+1j3 x*(x-l)* belongs to F as is readily seen. Hence, the field S = k (1) of a11 rational functions of 1 Will belong to F. We contend: F = S and (E/F) = 6. Indeed, from Theorem 13 we obtain (E/F) 1 6. Since S C F it suffices to prove (E/S) <- 6. Now E = S(x). It is thus sufficient to find some 6-th degree equation with coefficients in S satisfied by x. The following one is obviously satisfied; (x2 - x+1)3 -1.x2(x-1)2 = 0. The reader Will find the study of these fields a profitable exer- cise. At a later occasion he Will be able to derive a11 intermediate fields. 2) LetkbeafieldandE = k(x,,x2,...,x,)thefieldofall rational functions of n variables x1, x2, . . . , xn. If (vi , v2, . . . , Vu ) is a permutation of (1,2, . . . , n) we replace in each function f (x 1, x *, . . . , xn) of E the variable x, by x x2 by x xn by x1/ . The mapping of E Vl ’ 1/2’. . ’ n onto itself obtained in this way is obviously an automorphism and we may construct n ! automorphisms in this fashion (including the identity).

Let F be the fixed point field, that is, the set of a11 so-called

“symmetric functions.” Theorem 13 shows that (E/F) >- n ! . Let us in- troduce the polynomial: (3) f(t) = (t-xr)(t-x2). . . (t-x,) = t” + a itn-i + . + an 40 wherear = - (x1 + x2 + . . . + x,); a2 .: + (x1x2 + x1x3 + . . . + xn-rxn) and more generally ai is ( - 1)’ times the sum of a11 products of i differ- erent variables of the set x1, x2, . . . , xn. The functions a,, a*, . . . , an are called the elementary symmetric functions and the field S = k(a,,a,,..., an ) of a11 rational functions of a,, a2, . . . , ati is obviously a part of F. Should we suceed in proving ( E/S ) <- n ! we would have shown S = F and (E/F) = n ! . We construct to this effect the following tower of fields:

S = S, c SnT1 c Snm2 . . . c S, c S, = E by the definition

(4) sn = s; si = s(xi+l ‘xi+*‘...‘xn) = si+l (xi+l 1.

It would be sufficient to prove ( Si-r/S, ) <- i or that the generator xi for Si-r out of S satisfies an equation of degree i with coefficients in S, . Such an equation is easily constructed. Put Fi+, (t) (5) Fi (t> = f(t) (t-xi+l )(t-Xi+* ). . . (t-x,) = (t-xi+l ) and Fn ( t ) = f ( t ). Performing the division we see that Fi (t ) is a polynomial in t of degree i whose highest coefficient is 1 and whose coefficients are polynomials in the variables a1 , azF . . . , a,, and xi+r , xi+2 , . . . , x,. Only integers enter as coefficients in these expressions. Now xi is obviously a root of Fi (t ) = 0.

Nowletg(x,,x,,..., xn) be a polynomial in x1,x2,. . . , xn. Since Fr ( xr ) = 0 is of first degree in xr , we cari express x, as a polynomial of the a, and of x2, x3, . . . , xn . We introduce this expression ing(x,,x,,..., xn). Since F, (x2 ) = 0 we cari express x2 or higher 41 powers as polynomials in x3, . . . , xn and the ai. Since F, ( xg) = 0 we cari express xi and higher powers as polynomials of x4, x5,. . . , x,, and the ai. Introducing these expressions in g( xi, x2,. . . , xn) we see that we cari express it as a polynomial in the xr, and the ar, such that the degree in xi is below i. SO g( xi, x2, . . . , xn) is a linear combination of the following n ! terms:

Vl V2 Un x2 . . . xn (6) x1 where each vi 5 i - 1. The coefficients of these terms are polynomials in the ai . Since the expressions (6) are linearly independent in S (this is our previous result), the expression is unique. This is a generalization of the theorem of symmetric functions in its usual form. The latter says that a symmetric polynomial cari be written. as a polynomial in ai, a,, . . . , a,,. Indeed, if g(x,, . . . ,x,) is symmetric we have already an expression as linear combination of the terms (,6) where only the term 1 corresponding to vi = v2 = . . . = v,, = 0 has a coefficient # 0 in S, namely, g( xi,. . . , x,.,). SO g( xi, x2,. . . , xn) is a polynomial in a,, a2,. . . , a,. Hut our theorem gives an expression of any polynomial, symmetric or not.

H. Normal Extensions. An extension field E of a field F is called a normal extension if the group G of automorphisms of E which leave F fixed has F for its fixed field, and (E/F) is finite. Mthough the result in Theorem 13 cannot be sharpened in general, 42 there is one case in which the equality sign Will always occur, namely, in the case in which CT~, 02, . . . , an is a set of automorphisms which form a group. We prove

THEOREM 14. If ol> g2>. . . , o,, is a group of automorphisms of a field E and if F is the fixed field of g1 ,u*,. . . , on, then (E/F) = n. -- Ifa,,o,,...,un is a group, then the identity occurs, say, u1 = 1. The fixed field consists of those elements x which are not moved by any of the U’S, i.e., ai(x) = x, i = 1,2, . . . n. Suppose that (E/F ) > n.

Then there exist n + 1 elements al, a*, . . . , a,,, of E which are linearly independent with respect to F. By Theorem 1, there exists a non-trivial solution in E to the system of equations x1 q (a1 > + x2 q (a, > + . . . + xn+, q (a,,, > = 0

x1 u2 (a1 > + x2 o2 (a2 > + . . . + xn+l a2 ( a,+l > = 0 (’ > ...... xlun(al) + x2un(a2) + . . . + xn+lun(an+l> = 0

We not.e that the solution cannot lie in F, otherwise, since u1 is the identity, the first equation would be a dependence between a,, . . . , a n+l . Among a11 non-trivial solutions x1, x2, . . . , x,+~ we choose one which has the least number of elements different from 0. We may sup- pose this solution to be al, a*, . . . , a,, 0, . . . , 0, where the first r terms are different from 0. Moreover, r # 1 because a, u1 (a1 ) = 0 implies a1 = 0 since u1 (a, ) = a1 f 0. Also, we may suppose ar = 1, since if we multiply the given solution by a;’ we obtain a new solution in which the r-th term is 1. Thus, we have 43

(*> alUi(al > + a2Ui(a2 > + a-. + ar.,ui(a,_l > + oi(ar> = 0 for i = 1,2, . . . , n. Since ai, . . . , ar_i cannot a11 belong to F, one of these, say a 1, is in E but not in F. There is an automorphism ok for which o,,( a, ) # a1 . If we use the fact that oi, 02, . . . , o,, form a group, we see ok. gi, ok* 02,. . . ,ak. an is a permutation of Oi,oz,. . - tu,.

Applying ok to the expressions in (*) we obtain

Ukk(a,).UkQj(al > + *f. + ~~(ar-l)-~~Qj(a,-l) + @kUj(ar) = 0 for j = ‘l,2,. . . , n, SO that from okaj = (T.1

(**> Uk(al >Ui(al> + . ..+ Uk(ar-l)Ui(a,-l > + Ui(a, > = 0 and if we subtract ( * * ) from ( * ) we have

[a, - uk(al >l. Qi(Q1 > + . .. + [ar-l - uk(arel >l~i(Qr-l) = 0 which is a non-trivial solution to the system (’ ) having fewer than r elements different from 0, contrary to the choice of r.

-Corollary 1. If F is the fixed field for the finite group G, then each automorphism u that leaves F fixed must belong to G.

(E/F) = order of G = n. Assume there is a u not in G. Then F would remain fixed under the n + 1 elements consisting of u and the elements of G, thus contradicting the corollary to Theorem 13.

Corollary 2. There are no two finite groups G, and G, with the same fixed field.

This follows immediately from Corollary 1.

If f(x) is a polynomial in F, then f(x) is called separable if its irreducible factors do not have repeated roots. If E is an extension of 44 the field F, the element a of E is called separable if it is root of a separable polynomial f(x) in F, and E is called a separable extension if each element of E is separable.

THEOREM 15. E is a normal extension of F if and only if E is the-- splitting field of a separable polynomial p(x) in F. Sufficiency. Under the assumption that E splits p (x) we prove that E is a normal extension of F.

If a11 roots of p(x) are in F, then our proposition is trivial, since then E = F and only the unit automorphism leaves F fixed.

Let us suppose p(x) has n > 1 roots in E but not in F. We make the inductive assumption that for a11 pairs of fields with fewer than n roots of p(x) outside of F our proposition holds.

Let p(x) = pr(x).p,(x).. . . . pr( x) be a factorization of p(x) into irreducible factors. We may suppose one of these to have a degree greater than one, for otherwise p(x) would split in F. Suppose deg p,(x) = s > 1. Let ai be a root of p,(x). Then (F(a, )/F) = deg p,(x) = s.

If we consider F (ai ) as the new ground field, fewer roots of p( x) than n are outside. From the fact that p(x) lies in F(a, ) and E is a split- ting field of p(x) over F( a1 ), it follows by our inductive assumption that E is a normal extension of F(a, ). Thus, each element in E which is not in F(a, ) is moved by at least one automorphism which leaves

F(a, ) fixed.

p (x) being separable, the roots ai ,aZ, . . . , as of pi (x) are a distinct elements of E. By Theorem 8 there exist isomorphisms 45

Ul>ff2>. f *, os mappingF(a,)onF(a,), F(a,),...,F(a,), respect.ively, which are each the identity on F and map or on a1,a2,...,as respectively. We now apply Theorem 10. E is a splitting field of p(x) in F(a, ) and is also a splitting field of p(x) in F(ai ). Hence, the isomorphism oi, which makes p( x ) in F ( a1 ) correspond to the same p(x) in F( ai ), cari be extended to an isomorphic mapping of

E onto :E, that is, to an automorphism of E that we denote again by oi. Hence, u1,02,. . . , os are automorphisms of E that leave F fixed and map a1 ont0 a1,a2,. . . a,. Now let 8 be an element that remains fixed under a11 automor- phisms of E that leave F fixed. We know already that it is in F (a 1 ) and hence has the form

8 = CO + CIaI + C2 a: + . . . + csml as-l where the ci are in F. If we apply ui to this equation we get, since

Ui(0) = 8:

8 = C, + CIai + C2af + . . . + Csml a:-’

The polynomial es-r xs-1 + csd2x s-z + . . . + crx + (c Cl - 0) has therefore the s distinct roots or, 02, . . . , as. These are more than its degree. SO a11 coefficients of it must vanish, among them c 0 - 8. This shows 8 in F. Necessity. If E is a normal extension of F, then E is spiitting field of a separable polynomial p(x). We first prove the

-~Lemma. If E is a normal extension of F, then E is a separable extension of F. Moreover any element of E is a root ofan equation over

F which splits completely in E. 46

Let (5, , u2, . . . , o, be the group G of automorphisms of E whose fixed field is F. Let a be an element of E, and let Q, a2, a3, . . . ,cr be the set of distinct elements in the sequence oi( a), O~(U), . . . , os(c). Since G is a group, Uj(Qi) = Uj(c7,(U)) = ajo&) = a,(u) = a,. Therefore, the elements u,u2, . . . , cr are permuted by the automorphisms of G. The coefficients of the polynomial f(x) = (~-a)( X-U *). . . (xa,) are left fixed by each automorphism of G, since in its factored form the factors of f(x) are only permuted. Since the only elements of E which are left fixed by a11 the automorphisms of G belong to F, f(x) is a polynomial in F. If g(x) is a polynomial in F which also has a as root, then applying the automorphisms of G to the expression g (u ) = 0 we obtain g(a,) = 0, SO that the degree of g(x) -> s. Hence f(x) is irre- ducible, and the lemma is established.

TO complete the proof of the theorem, let or, 02,. . . , ut be a gen- erating system for the vector space E over F. Let fi(x) be the separable polynomial having oi as a root. Then E is the splitting field of p(x) = f,(x).f,(x).....f,(x). If f(x) is a polynomial in a field F, and E the splitting field of f (x ), then we shall cal1 the group of automorphisms of E over F the

-~group of the equation f(x) = 0. We corne now to a theorem known in algebra as the Fundamental Theorem of Galois Theory which gives the relation between the structure of a splitting field and its group of automorphisms. THEOREM 16. (Fundamental Theorem). If p(x) is a separable polynomial in a field F, and G the group of the equation p(x) = 0 where--- E is the 47 splitting field of p(x), then: (1) Each intermediate field, B,is the fixed field for a subgroup G, of G, and distinct subgroups have dis- tinct fixed fields. We say B and G, “belong” to each other. (2) The intermediate field B is a normal extension of F if a,nd only if the sub- group G, is a normal subgroup of G. In this case the group of automor- phisms of B which leaves F fixed is isomorphic to the factor group

(G/G ).- (3) For each intermediate field B, we have (B/F) = index of G, and (E/B) = order of G,.

The first part of the theorem cornes from the observation that E is splitting field for p(x) when p(x) is taken to be in any intermediate field. Hence, E is a normal extension of each intermediate field B, SO that B is the fixed field of the subgroup of G consisting of the automor- phisms which leave B fixed. That distinct subgroups have distinct fixed fields is stated in Corollary 2 to Theorem 14.

Let B be any intermediate field. Since B is the fixed field for the subgroup G, of G, by Theorem 14 we have (E/B) = order of G,. Let us cal1 o(G) the order of a group G and i(G) its index. Then o(G) =: o(Ga But (E/F) = o(G), and (E/F) = (E/B)*(B/F) from which (B/F) = i (G, ), which proves the third part of the theorem.

The number i( G, ) is equal to the number of left cosets of G,. The elements of G, being automorphisms of E, are isomorphisms of B; that is, they map B isomorphically into some other subfield of E and are the identity on F. The elements of G in any one coset of G, map B in the same way. For let U. a1 and o. o2 be two elements of the coset uG,. Since or and o2 leave B fixed, for each a in B 48 we have (ror( a ) = a( cz ) = oo2( a ). Elements of different cosets give different isomorphisms, for if o and r give the same isomorphism, o(a) = r(c) for each a in B, then o-‘r(a) = a for each a in B. Hence, dT == al> where or is an element of G,. But then T = oo, and rG, := oorGs= aG, SO that o and r belong to the same coset. Each isomorphism of B which is the identity on F is given by an automorphism belonging to G. For let Q be an isomorphism mapping B on B’ and the identity on F. Then under c, p(x) corresponds to p(x), and E is the splitting field of p(x) in B and of p(x) in B ’ . By Theorem 10, a cari be extended to an automorphism o’ of E, and since 0’ leaves F fixed it belongs to G. Therefore, the number of distinct isomorphisms of B is equal to the number of cosets of G, and is there- fore equal to (B/F). The field aB onto which g maps B has obviously oGso-1 as cor- responding group, since the elements of aB are left invariant by precisely this group. If 13 is a normal extension of F, the number of distinct automor- phisms of B which leave F fixed is (B/F) by Theorem 14. Conversely, if the number of automorphisms is (B/F) then B is a normal extension, because if F’ is the fixed field of a11 these automorphisms, then F C F’ (1 B, and by Theorem 14, (B/F ‘) is equal to the number of automorphisms in the group, hence (B/F ‘) = (B/F). From ( B/F) = (B/F’)(F’/F) we have (F’/F) = 1 or F = F’. Thus, B is a normal extension of F if and only if the number of automorphisms of B is (B/F). B is a normal extension of F if and only if each isomorphism of B into E is an automorphism of B. This follows from the fact that each of the above conditions are equivalent to the assertion that there are 49 the same numberof isomorphisms and automorphisms. Since, for each u, B = aB is equivalent to oG,o -* C G,, we cari finally say that B is a normal extension of F and only if G, is a normal subgroup of G.

As we have shown, each isomorphism of B is described by the effect of the elements of some left coset of G,. If B is a normal exten- sion these isomorphisms are a11 automorphisms, but in this case the cosets are elements of the factor group (G/G, ). Thus, each automor- phism of B corresponds uniquely to an element of ( G/G, ) and con- versely. Since multiplication in ( G/G, ) is obtained by iterating the mappings, the correspondence is an isomorphism between (G/G, ) and the group of automorphisms of B which leave F fixed. This completes the proof of Theorem 16.

1. Finite Fields.

It is frequently necessary to know the nature of a finite subset of a field which under multiplication in the field is a group. The answer to this question is particularly simple.

-THEOREM 17. If S is a finite subset (f 0 ) of a field F which

~~is a group under multiplication in F, then S is a cyclic group. The proof is based on the following lemmas for abelian groups.

-L,emma 1. If in an abelian group A and B are two elements of

~-orders a and b, and if c is the least common multiple of a and b, then

~~there is an element C of order c in the group. 50

Proof: (a) If a and b are relatively prime, C = AB has the re- quired order ab. The order of C a = B” is b and therefore c is divisible by b. Similarly it is divisible by a. Since Ceb = 1 it follows c = ab.

(b) If d is a divisor of a, we cari find in the group an element of order d. Indeed Aa/d is this element.

(c) Now let us consider the general case. Let pr, pz, . . . , p, be the prime numbers dividing either a or b and let

a = p,“1 pz n2 . . . prnr

b = pimlpzm;, . ..prm’.

Cal1 ti the larger of the two numbers ni and 1. Then

t c = p* t1 p2 t2 . . . prr*.

According to (b) we cari find in the group an element of order pni and one of order pi?. Thus there is one of order pi ti. Part (a) shows that the product of these elements Will have the desired order c.

--Lemma 2. If there is an element C in an abelian group whose

--order c is maximal (as is always the case if the group is finite) then c

--is divisible by the order a of every element A in the group; hence c --x = 1 is satisfied by each element in the group. Proof: If a does not divide c, the greatest common multiple of a and c would be larger than c and we could find an element of that order, thus contradicting the choice of c. We now prove Theorem 17. Let n be the order of S and r the largest order occuring in S. Then x’ - 1 = 0 is satisfied for a11 ele- 51 ments of S. Since this polynomial of degree r in the field cannot have more than r roots, it follows that r -> n. On the other hand r -< n be- cause the order of each element divides n. S is therefore a cyclic group consisting of l,~, ~2,. . . , c”-I where 6” = 1.

Theorem 17 could also have been based on the decomposition theorem for abelian groups having a finite number of generators. Since

this theorem Will be needed later, we interpolate a proof of it here.

Let G be an abelian group, with group operation written as +.

The element g,, . . . , g, Will be said to generate G if each element g of

G cari be written as sum of multiples of g,, . . . , g,, g = n,g, + . . . + nkgk.

If no set of fewer than k elements generate G, then g,, . . . , G Will be called a minimal generating system. Any group having a finite genera-

ting system admits a minimal generating system. In particular, a finite

group al.ways admits a minimal generating system.

From the identity nl( g, + mg,) + (n2 - n,m)g2 = n,g, + n2g,

it follows that if g,, g,, . . . , g, generate G, also g, + mg,,

g,, * . . >g, generate G.

An equation m,g, + m,g, + . . . + m,g, = 0 Will be called a re-

lation between the generators, and ml, . . . , mk Will be called the co-

efficients in the relation.

We shall say that the abelian group G is the direct product of its

subgroups G,, G,, . . . , G, if each g E G is uniquely representable as a sumg = x1 + x2 + . . . + x,,wherexi E Gi,i = l,..., k.. 52

-Decomposition Theorem. Each abelian group having a finite num- ~~ber of generators is the direct product of cyclic subgroups G,, . . . , G, where the order of Gi divides the order of Gi+i, i = 1, . . . , n-l and n is ~- the number of elements in a minimal generating system. ( Gr, Gr+i , . . . , Gn may each be infinite, in which case, to be precise, O(Gi)lO(Gi+,)fori = 1,2,...,r-2). We assume the theorem true for a11 groups having minimal genera- ting systems of k-l elements. If n = 1 the group is cyclic and the theorem trivial. Now suppose G is an abelian group having a minimal generating system of k elements. If no minimal generating system satis- fies a non-trivial relation, then let g,, g,, . . . , g, be a minimal generating system and G,,G,, . . . , G, be the cyclic groups generated by them. For each g 6 G, g = n,g, + . . . + nkgk where the expression is uniqu.e; otherwise we should obtain a relation. Thus the theorem would be true. Assume now that some non-trivial relations hold for some mini- mal generating systems. Among a11 relations between minimal genera-

ting systems, let

(1) m,g, + . . . + mkg, = 0 be a relation in which the smallest positive coefficient occurs. After an eventual reordering of the generators we cari suppose this coefficient to be mi. In any other relation between g,, . . . , g,.

(2) ni g, + . . . + nkgk = 0

we must have mi/ni. Otherwise n = qmi + r, 0 < r < mi and q times 1 relation (1) subtracted from relation (2) would yield a relation with a coefficient r < mi. Also in relation (1) we must have m,/m,, i = 2,. . . , k. 53

For suppose mi does not divide one coefficient, say m, . Then In the generating system m2 = qm, + r, 0 < r < mr. g, + g,, k$‘...> g, we should have a relation mi( g, + qg,) + rg, + m,g, + . . . + mkq, = 0 where the coefficient r contradicts the choice of mi. Hence m2 = q2m1, m3 = q,m,, . . . , mk = q,m,. The system & = g, + q,g, + . . . + qkgk, g,, . . . , g, is minimal gen- erating, and m,gr = 0. In any relation 0 = n,Fi + n2g2 + . . . + nkgk since mr is a coefficient in a relation between gi, g,, . . . , g, our pre- vious argument yields mr / nr , and hence nr gr = 0.

Let G’ be the subgroup of G generated by g,, . . . , g, and G, the cyclic group of order m, generated by gr . Then G is the direct product of G, and G’ . Each element g of G cari be written

g = nigi + n2g2 + . . . + nkg, = nrgr + g’.

The representation is unique, since n,g, + g’ = nr’gl + g” implies the relation (nr - nr’)g, + (g’ - g”) = 0, hence

(nl - ni )E, = 0, SO that nrgr = n;gi and also g’ = g”. E3y our inductive hypothesis, G ’ is the direct product of k-l cyclic groups generated by elements g2, ES, . . . , gk whose respective orders t,, . . . , t, satisfy ti / ti+r , i = 2, . . . , k-l. The preceding argu- ment applied to the generators gr, g2, . . . , g, yields m, j t,, from which the theorem follows.

I3y a finite field is meant one having only a finite number of elements.

-~~Corollary. The non-zero elements of a finite field form a cyclic group. If a is an element of a field F, let us denote the n-fold of a, i.e., 54 the element of F obtained by adding a to itself n times, by na. It is ob- vious that n.(m.a) = (nm).a and(n.a)(m.b) = nmeab. If for one element a f 0, there is an integer n such that na a = 0 then n. b = 0 foreachbinF,sincen.b=(n.a)(a-‘b)=O.a-’b=O..Ifthereisa positive integer p such that p. a = 0 for each a in F, and if p is the smallest integer with this property, then F is said to have the charac-

--teristic p. If no such positive integer exists then we say F has charac- teristic 0. The characteristic of a field is always a prime number. for if p = r.s thenpa = rs.a = r.(s.a). However, s.a = b # Oif a f 0 and r b + 0 since both r and s are less than p, SO that pa f 0 contrary to the definition of the characteristic. If na = 0 for a f 0, then p divides n, for n = qp + r where 0 -< r < p and na = (qp + r)a = qpa + ra. Hence na = 0 implies ra = 0 and from the definition of the characteristic since r < p, we must have r = 0. If F is a finite field having q elements and E an extension of F such that (E/F) = n, then E has q” elements. For if or, 02,. . . ,W n is a basis of E over F, each element of E cari be uniquely represented as a linear combination xiwr + xZoZ + . . . + x,w, where the xi belong to F. Since each xi cari assume q values in F, there are qn distinct possi- ble choices of x1, . . . , xn and hence qn distinct elements of E. E is finite,, hence, there is an element a of E SO that E = F(a). (The non- zero elements of E form a cyclic group generated by a). IfwedenotebyP 5 [0,1,2 ,..., p-l] the set of multiples of the unit element in a field F of characteristic p, then P is a subfield of F having p distinct elements. In fact, P is isomorphic to the field of integers reduced mod p. If F is a finite field, then the degree of F over 55

P is finite, say (F/P) = n, and F contains p* elements. In other

words, the order of any finite field is a power of its characteristic.

If F and F’ are two finite fields having the same order q, then

by the preceding, they have the same characteristic since q is a power

of the characteristic. The multiples of the unit in F and FI form two

fields P and P’ which are isomorphic.

The non-zero elements of F and F’ form a group of order q-l

and, therefore, satisfy the equation xq-l - 1 = 0. The fields F and FI

are splitting fields of the equation x q-l = 1 considered as lying in P

and P’ respectively. By Theorem 10, the isomorphism between P and

P ’ cari be extended to an isomorphism between F and F ’ . We have thus proved

THEOREM 18. Two finite fields having the same number of ele-

~-ments are isomorphic. gifferentiation. If f(x) = ao + six + . . . + anxn is a poly-

nomial in a field F, then we define f’ = a, + 2a,x + . . . + nanx”-i .

The reader may readily verify that for each pair of polynomials f and

g we have

(f + g)’ = f’ + g’

(f g>’ = fg’ + gf’ (f^)’ = nf”-‘. f’

-THEOREM 19. The polynomial f has repeated roots if and only if in the splitting field E the polynomials f and f’ have a common

root. This condition is eauivalent to the assertion that f and f’ have a 56

~-common factor of degree greater than 0 in F. If a is a root of multiplicity k of f(x) then f = (x-a )kQ (x) where

Q(a) f 0. This gives f ’ = ( x-a )kQ ’ ( x ) + k ( x-a ) k-1 Q ( x ) = ( x-a ) k-1 [ ( x-a ) Q’ ( x ) + kQ ( x ) 1.

If k > 1, then a is a root of f’ of multiplicity at least k-l. If k = 1, then f’(x) = Q(x) + (x-a)Q’(x) and f’(a) = Q(a) f; 0. Thus, f and f’ have a root a in common if and only if a is a root of f of multiplicity greater than 1. If f and f’ have a root a in common then the irreducible polynomial in F having a as root divides both f and f’ . Conversely, any root of a factor common to both f and f’ is a root of f and f ’ .

-~Corollary. If F is a field of characteristic 0 then each irreducible ~~polynomial in F is separable. Suppose to the contrary that the irreducible polynomial f(x) has a root a of multiplicity greater than 1. Then, f ’ (x) is a polynomial which is not identically zero (its leading coefficient is a multiple of the leading coefficient of f(x) and is not zero since the characteristic is 0) and of degree 1 less than the degree of f(x). But a is also a root of f’ (x) which contradicts the irreducibility of f(x).

J. Roots-- of Unity. If F is a field having any characteristic p, and E the splitting field of the polynomial x D - 1 where p does not divide n, then we shall refer to E as the field generated out of F by the adjunction of a

~~primitive nth root of unity. The polynomial x” ~ 1 does not have repeated roots in E, since its derivative, nx”-’ , has only the root 0 and has, therefore, no roots 57 in common with x n - 1. Thus, E is a normal extension cf F. IfE,,t* ,..., Cn are the roots of x n - 1 in E, they form a group under multiplication and by Theorem 17 this group Will be cyclic. If 1 ,C,C2,a. .,c n-1 are the elements of the group, we shall cal1 c a primi- tive n th root of unity. The smallest power of E which is 1 is the n th.

-~THEOREM 20. If E is the field generated from F by a primitive ~~nth root of unity, then the group G of E over F is abelian for any n and ~-cyclic if n is a prime number. We have E = F(É), since the roots of x” - 1 are powers of 6. Thus, if o and r are distinct elements of G, o(c) f r(e). But ~(6) is a root of xn - 1 and, hence, a power of 6. Thus, o(c) = c ““where no is an integer 1 5 no < n. Moreover, ru(~) = r( r”o) = (r(t)) “o = cnr “u = or(~). Thus, nor = snr mod n. Thus, the mapping of o on no is a homomorphism of G into a multiplicative subgroup of the integers mod n. Since r f o implies T(E) f o(c), it follows that r f o implies

“a f nr mod n. Hence, the homomorphism is an isomorphism. If n is a prime number, the multiplicative group of numbers forms a cyclic group.

K. Noether Equations. If E is a field, and G = ( CJ, r, . . .) a group of automorphisms of E, any set of elements x,, xr , . . . in E Will be said to provide a solution to

Noether’s~~ equations if xo . D (xr ) = xor for each o and r in G. If one element x0 = 0 then xr = 0 for each r 6 G. As T traces G, or assumes a11 values in G, and in the above equation xor = 0 when xo = 0. Thus, in any solution of the Noether equations no element xo = 0 unless the solution is completely trivial. We shall assume in the sequel that the 58 trivial solution has been excluded. THEOREM 21. The system xo, xr , . . . is a solution to Noether’s - equations if and only if there exists an element a in E, such that

=XfJ CL/(~( a ) for each o. For any a, it is clear that xo = a/o( a) is a solution to the equations, since

a/o(a)~o(a/r(a>) = a/a(a>*o(a)/ui-(a) = a/o7(a). Conversely, let x0, xr , . . . be a non-trivial solution. Since the automorphisms o, T , . . . are distinct they are linearly independent, and the equation x0 ‘U(Z) + XrT(Z) + . . . = 0 does not hold identically. Hence, there is an element a in E such that xo.cr(a) + x,r(a) + . . . = a f 0. Applying u to a gives o(a) = T,CG a(x,).m(a>. Multiplying by ~0 gives

xu.u(u) = C xoo(x,).or(a>. 7CG Replacing xc. c( xr ) by b and noting that or assumes a11 values in G when r does, we have

Xrlr . o(a) = C x,r(a) = a TEG SO that xu = a/o(a). A solution to the Noether equations defines a mapping C of G into E, namely, C(a) = x,,. If F is the fixed field of G, and the ele- ments xc lie in F, then C is a character of G. For C(OT)= xm = x,.a(x,) = x0x7= C(a).C(r)sinceo(x,) = x,if xr c F. Conversely, each character C of G in F provides a solution 59 to the Noether equations. Cal1 C(o) = xo. Then, since xr 6 F, we have a( xr) = xr . Thus, xu .a(s:,) == xoexr = C(~)*C(T) = C(or) = xor. We thereforehave, by combining this with Theorem 21, THEOREM 22. If G is the group of the normal field E over F, then for each character C of G into F there exists an element Q in E

~~such that C(a) = a/o(a) and, conversely, if a/o( a) is in F for each

(T, then C ((7) = a/o( a ) is a character of G. If r is the least common

~~multiple of the orders of elements of G, then a’ c F. We have already shown a11 but the last sentence of Theorem 22.

TO prove this we need only show o(ar) = a’ for each o E G. But ar/a(ar) = (a/a(a))= = (C(o))r = C(or) = C(1) = 1.

L. Kummer’s Fields. If F contains a primitive nth root of unity, any splitting field E of a polynomial (x” - a,)(x” - a*). . .(x” - ar) where ai 6 F for i = 1,2,. . . , r Will be called a Kummer extension of F, or more briefly, a ----.--zKummer field If a field F contains a primitive nth root of unity, the number n is not divisible by the characteristic of F. Suppose, to the contrary, F has characteristic p and n = qp. Then yn - 1 = (y - 1 )n since in the expansion of (y - 1 )P each coefficient other than the first and last is divisible by p and therefore is a multiple of the p-fold of the unit of F and thus is equal to 0. Therefore x” - 1 = (x q)P - 1 = (x4 - 1 )n and x” - 1 cannot have more than q distinct roots. But we assumed that F has a primitive n th root of unity and 1, 6,~ 2,. . . , E n-1 would be 60 n distinct roots of x” - 1. It follows that n is not divisible by the characteristic of F. For a Kummer field E, none of the factors

X” - ai, ai f 0 has repeated roots since the derivative, nx”-l , has only the root 0 and has therefore no roots in common with xn - ai.

Therefore, the irreducible factors of x” - a, are separable, SO that -E is a normal extension of F.

Let ai be a root of x” - ai in E. If ci, c2, . . . , tn are the n dis- tinct nth roots of unity in F, then ai+,, aic2, . . . , a,~,, Will be n distinct

roots of x” - ai, and hence Will be the roots of x” - a,, SO that

E = F(a,,a,,..., ar). Let o and r be two automorphisms in the group G of E over F. For each ai, both o and r map ai on some other root of x” - ai. Thus r(ai) = ciTai and o(ai) = cio ai where Q, and tir are nth roots of unity in the basic field F. It follows that

r(o(ai)) = T(tiaai) = ciuT = tiocirai = o(T(ai). Since oand r are commutative over the generators of E, they commute over each ele-

ment of E:. Hence, G is commutative. If ~7 c G, then o(ai) =

ciGai, a'(a,) = eio 2ai, etc. Thus, oni = ai for ni such that “i %J = 1. Since the order of an n th root of unity is a divisor of n, we have ni a divisor of n and the least common multiple m of nr, n2,. . . , nr

is a divisor of n. Since o ya,) = ai for i = 1,2,. . . , r it follows that m is the order of CJ. Hence, the order of each element of G is a divisor of n and, therefore, the least common multiple r of the orders of the ele- ments of G is a divisor of n. If c is a primitive nth root of unity, then en/’ is a primitive r th root of unity. These remarks cari be summarized in the following. 61

--THEOREM 23. If E is a Kummer field, i.e., a splitting field of

p(x)~- = (x” - a,)(x” - az). . .(x” - ar) whete a, lie in F, and F contains~~ a primitive nth root of unity, then: (a) E is a normal extension

~-of F; (b) the group G of E over F is abelian, (c) the least common multi- ~~ple of the orders of the elements of G is a divisor of n. ---Corollary. If E is the splitting field of xp - a, and F contains a

~~primitive pth root of unity where p is a prime number, then either E = F and xp -- a is split in F, or xp - a is irreducible and the group of E

over F is cyclic of order p. The order of each element of G is, by Theorem 23, a divisor of p and, hence, if the element is not the unit its order must be p. If Q is a

root of xp - a, then a,ey, . . . ,cp-lc are a11 the roots of xP - a SO that

F(a ) = E and ( E/F) < -p . Hence, the order of G does not exceed p SO that if G has one element different from the unit, it and its powers must constitute a11 of G. Since G has p distinct elements and their behavior

is determined by their effect on a, then a must have p distinct images.

Hence, the irreducible equation in F for a must be of degree p and is therefore xp - a = 0. The properties (a), (b) and (c) in Theorem 23 actually characterize Kummer fields. Let us suppose that E is a normal extension of a field F, whose group G over F is abelian. Let us further assume that F contains a primitive r th root of unity where r is the least common multiple of the orders of elements of G. The group of characters X of G into the group of r th roots of 62 unity is isomorphic to G. Moreover, to each cr 6 G, if 0 f 1, there exists a character C 6 X such that C (0) f 1. Write G as the direct product of the cyclic groups G,, G,, . . . , G, of orders mi / m2 / . . . / m,. Each u E G may be written D = 0rVl 0z v2 . . .0rVt . Cal1 Ci the character sending cri into ci, a primitive m.Ith root of unity and 5 into 1 for j f i. Let C be any character. C(oi) = cri, then we have C = CFi . C 2 i. . . C,’ i. Conversely, C P1 . . . Ct Pr defines a character. Since the order of Ci is 1 mi, the character group X of G is isomorphic to G. If 0 f 1, then in 9 1.‘2 Vt u=o 1 u2 . ..u t at least one vi, say vr , is not divisible by mi. Thus Cr( u) = civl f 1.

Let A denote the set of those non-zero elements cz of E for which a’ 6 F and let Fi denote the non-zero elements of F. It is obvious that A is a multiplicative group and that Fi is a subgroup of A. Let A’ de- note the set of rth powers of elements in A and Fi the set of r th powers of elements of Fi. The following theorem provides in most applications a convenient method for computing the group G.

THEOREM 24. The factor groups (A/F, ) and (A’/F ;) are iso- morphic to each other and to the groups G and X.

We map A on A’ by making a 6 A correspond to a’ 6 A’. If a’ t Fi, where a 6 F, then b c A is mapped on ar if and only if br = a=, that is, if b is a solution to the equation x’ - a’ = 0. But a, ca, c2a,. . . , cr-’ a are distinct solutions to this equation and since 6 and a belong to Fi, it follows that b must be one of these elements and must belong to Fi.

Thus, the inverse set in A of the subgroup Fi of A’ is Fi , SO that the

factor groups (A/F, ) and (A’/F; ) are isomorphic. 63

If a is an element of A, then (a/o(~))~ = ar/o(a*) = 1. Hence, a/o(a) is an rth root of unity and lies in F 1. By Theorem 22, ~/a (a )

defines a character C (0) of G in F. We map a on the corresponding

character C. Each character C is by Theorem 22, image of some a.

Moreover, u . a ’ is mapped on the character C * (0) = a.a’/o(a.a’) = a.a’/o(a>-a(a’> = C(o>.C’(a> = C.C’(u), SO that the mapping is homomorphism. The kernel of this homomorphism

is the set of those elements a for which a/a(a) = 1 for each o, hence is Fi. It follows, therefore, that (A/F, ) is isomorphic to X and hence

also to G. In particular, (A/F, ) is a finite group.

We now prove the equivalence between Kummer fields and fields

satisfying (a), (b) and (c) of Theorem 23.

THEOREM 25. If E is an extension field over F, then E is a

Kummet field if and only if E is normal, its group G is abelian and F

contains a primitive rth root 6 of unity where r is the least common

multinle of the orders of the elements of G.

The necessity is already contained in Theorem 23.. We prove the

sufficiency. Out of the group A, let a,F,,a,F,, . . . ,a,F, be the cosets

of Fi. Since ai E A, we have af = ai 6 F. Thus, ai is a root of the

equation x’ - ai = 0 and since coi, c2ai, . . . , c'-*ai are also roots,

X’ - ai must split in E. We prove that E is the splitting field of

(x’ - a1 )(x’ - a2 ). . . (x’ - a,) which Will complete the proof of the

theorem. TO this end it suffices to show that F ( ai, a2, . . . , ut) = E. 64

Suppose that F(a,,a*, . . . ,a,) f E. Then F(a,, . . . ,a,) is an intermediate field between F and E, and since E is normal over

F(a,,... , ut) there exists an automorphism o 6 G, D f 1, which leavesF(a,,..., a,) fixed. There exists a character C of G for which

C(a) f 1. Finally, there exists an element a in E such that

C(o) = a/a(a) f 1. But a* E Fi by Theorem 22, hence a 6 A.

Moreover, A C F(a,, . . . ,a, ) since a11 the cosets a,F, are contained in F(a,, . . . ,a,). Since F(a,, . .. ,a,) is by assumption left fixed by o, a(a) = a which contradicts a/o(a) f 1. It follows, therefore, that

F(a,,...,a,) = E.

-~Corollary. If E is a normal extension of F, of prime order p, and

--if F contains a primitive pth root of unity, then E is splitting field of

~~an irreducible polynomial xn - a in F.

E is generated by elements ai, . . . ,a,, where ay 6 F. Let ai be not in F.. Then xp - a is irreducible, for otherwise F(a 1 ) would be an intermediate field between F and E of degree less than p, and by the product theorem for the degrees, p would not be a prime number, con- trary to assumption. E = F ( a1 ) is the splitting field of xp - a.

M. Simple Extensions.

We consider the question of determining under what conditions an extension field is generated by a single element, called a primitive.

We prove the following

-THEOREM 26. A finite extension E of F is primitive over F if 65

~-and only if there are only a finite number of intermediate fields. (a) Let E = F(a) and cal1 f(x) = 0 the irreducible equation for a in F. Let B be an intermediate field and g(x) the irreducible equa- tion for Q in B. The coefficients of g(x) adjoined to F Will generate a field B ’ between F and B. g ( x ) is irreducible in B, hence also in B ’ .

Since 15 = B’(a) we see (E/B) = (E/B’). This proves B’ = B. SO B is uniquely determined by the polynomial g(x). But g(x) is a divisor of f(x), and there are only a finite number of possible divisors of f(x) in E. Hence there are only a finite number of possible B’s.

(b) Assume there are only a finite number of fields between E and F. Should F consist only of a finite number of elements, then E is generated by one element according to the Corollary on page 53. We may therefore assume F to contain an infinity of elements. We prove:

TO any two elements a, p there is a y in E such that F(a, /? ) = F (y). Let y -= a + ap with a in F but for the moment undetermined. Con- sider a11 the fields F(y) obtained in this way. Since we have an infinity of a’s at our disposal, we cari find two, say a, and a2, such that the corresponding y’s, yr = a + a,/3 and y2 = a + a,@, yield the same field F ( y1 ) = F ( y2 ). Since both y1 and y2 are in F ( y1 ), their difference (and therefore 6) is in this field. Consequently also

Y1 - a,P := a. SO F(a,P) C F(y,). Since F(y,) C F(a,B) our con- tention is proved. Select now I] in E in such a way that ( F ( T] )/F ) is as large as possible. Every element E of E mustbe in F( ‘7) or else we could fïnd an element 6 such that F(6) contains both 7 and 6. This proves E = F(n). 66

THEOREM 27. If E = F(a,, a2, . . . ,a,) is a finite extension of - the field F, and ai, a*, . . . ,a, are separable elements in E, then there exists-- a primitive 19 in E such that E = F (0). Proof: Let fi(x) be the irreducible equation of ai in F and let B be an extension of E that splits fi(x) fz( x) . . . f,(x). Then B is normal over F and contains, therefore, only a finite number of intermediate fields (as many as there are subgroups of G). SO the subfield E contains only a finite number of intermediate fields. Theorem 26 now compietes the proof.

N. Existence-~ of a Normal Basis. The following theorem is true for any field though we prove it only in the case that F contains an infinity of elements.

--THEOREM 28. If E is a normal extension of F and hi, oz, . . . , on -~are the elements of its group G, there is an element 8 in E such that

-~the n elements o,(t9),a,(ti), . . . , un( 0) are linearly independent with resuect to F.

According to Theorem 27 there is an a such that E = F(U). Let

f(x) be the equation for a, put o,(a) = ci,

and gi(x) = o,(g(x>> = (x-af,\~?ca.) 1 1 g i( x y) is a polynomial in E having ak as root for k f i and hence

(1) g,(x)g,(x) = 0 (mod f(x)) for i f k. In the equation

(2) g*(x) + g,(x) + . . . + g,(x) - 1 = 0 the left side is of degree at most n - 1. If (2) is true for n different values of x, the left side must be identically 0. Such n values are 67 al,a2,. . . ,a,, since gi( ai) = 1 and gk(ai ) = 0 for k f i. Multiplying (2) by gi( x) and using (1) shows:

(3) (g,(X))’ = gi(X) (mod f(x))- We next compute the determinant

(4) D(x) = loiok(g(x))l i,k = 1,2 ,..., n and prove D(x) f 0. If we square it by multiplying column by column and compute its value (mod f(x)) we get from (l), (2), (3) a determi- nant that has 1 in the diagonal and 0 elsewhere. SO (D(x))~ = 1 (mod f(x)). D (x 1) cari have only a finite number of roots in F. Avoiding them

we cari find a value a for x such that D(a) f 0. Now set 8 = g(a). Then the determinant

(5) l”iak(e)l f O’ C:onsider any linear relation

x1a,(8) + xp*(8> + . . . + ~,.,a,( 0) = 0 where the xi are in F. Apply- ing the automorphism oi to it would lead to n homogeneous equations for the n unknowns xi. (5) shows that xi = 0 and our theorem is proved.

0. ~~Theorem on Natural Irrationalities. L,et F be a field, p(x) a polynomial in F whose irreducible factors are separable, and let E be a splitting field for p(x). Let B be an arbi-

trary extension of F, and let us denote by EB the splitting field of p(x)

when p(x) is taken to lie in B. If al, . . . ,as are the roots of p(x) in EB, then F(a,, . . . ,as) is a subfield of EB which is readily seen to

form a splitting field for p(x) in F. By Theorem 10, E and F(a*, . . . , as ) 68 are isomorphic. There is therefore no loss of generality if in the sequel wetakeE = F(al,..., a,) and assume therefore that E is a subfield ofEB.Also,EB = B (a *,...> as).

Let us denote by E A B the intersection of E and B. It is readily seen that E n B is a field and is intermediate to F and E.

THEOREM 29. If G is the group of automorphisms of E over F,

--and H the group of EB over B, then H is isomorphic to the subgroup of G having E n B as its fixed field.

Each automorphism of EB over B simply permutes al, . . . , as in some fashion and leaves B, and hence also F, fixed. Since the ele- ments of EB are quotients of polynomial expressions in al, . . . , as with coefficients in B, the automorphism is completely determined by the permutation it effects on a 1, . . . , as. Thus, each automorphism of EB over B defines an automorphism of E = F ( al, . . . , as ) which leaves F fixed. Distinct automorphisms, since (x1,. . . , as belong to E, have different effects on E. Thus, the group H of EB ovet B cari be con- sidered as a subgroup of the group G of E over F. Each element of H leaves E n B fixed since it leaves even a11 of B fixed. Howevet, any element of E which is not in E n B is not in B, and hence would be moved by at least one automorphism of H. It follows that E r\ B is the fixed field of H.

--Corollaty. If, undet the conditions of Theorem 29, the gtoup G is of

--prime order, then either H = G or H consists of the unit element alone. 69

III APPLICATIONS

by A. N. Milgram

A. --Solvable Groups. Before proceeding with the applications we must discuss certain questions in the theory of groups. We shall assume several simple propo- sitions: (a) If N is a normal subgroup of the group G, then the mapping f(x) = xN is a homomorphism of G on the factor group G/N. f is called the natural homomorphism. (b) The image and the inverse image of a normal subgroup under a homomorphism is a normal subgroup. (c) If f is a homomorphism of the group G on G’ , then setting N’ = f(N), and defining the mapping g as g( xN ) = f(x) N ’ , we readily see that g is a homomorphism of the factor group G/N on the factor group G ‘/NI . Indeed, if N is the inverse image of N’ then g is an isomorphism. We now prove

-~~THEOREM 1. (Zassenhaus). If U and V are subgroups of G, u and v normal subgroups of U and V, respectively, then the following three factor groups are isomorphic: u (U nV) /II (U nv), v(UnV)/V(UnV), (unv)/(unv)(vnu). It is obvious that U n v is a normal subgroup of U n V. Let f be the natural mapping of U on V/u. Cal1 f(UnV) = H and f(Unv) = K. Then f-‘(H) = u(UnV) and f“(K) = u(Unv) from which it follows that u( UnV)/u( Unv) is isomorphic to H/K. If, however, we view f as defined only over U n V, then f-‘(K) = [un(UnV)](Unv) =

(unV)(Unv) SO that (UnV)/(unV)(Unv) is also isomorphic to H/K. 70

Thus the first and third of the above factor groups are isomorphic to

each other. Similarly, the second and third factor groups are isomorphic.

-Corollary 1. If H is a subgroup and N a normal subgroup of the group G, then H/HnN is isomorphic to HN/N, a subgroup of G/N.

Proof: Set G = U, N = u, H = V and the identity 1 = v in Theorem 1.

-Corollary 2. Under the conditions of Corollary 1, if G/N is

--abelian, SO also is H/HnN. Let us cal1 a group G solvable if it contains a sequence of sub-

groupsG = Go 1 G, I... 1 Gs = 1, each a normal subgroup of the

preceding, and with G,-r /Gi abelian.

-THEOREM 2. Any subgroup of a solvable group is solvable. For let H be a subgroup of G, and cal1 Hi = HnG,. Then that Hi-r/H, is

abelian follows from Corollary 2 above, where Gi_r, Gi and Hi_, play

the role of G, N and H.

-THEOREM 3. The homomorph of a solvable group is solvable. Let f(G) = G’ , and define G 1 = f ( Gi) where Gi belongs to a a sequence exhibiting the solvability of G. Then by (c) there exists a

homomorphism mappingGi_r/Gi on Gi_,/Gi . But the homomorphic image

of an abelian group is abelian SO that the groups GI exhibit the

solvability of G’ .

B. --Permutation Groups. Any one to one mapping of a set of n abjects on itself is called

a ---permutation. The iteration of two such mapping is called their product. 71

It may be readily verified that the set of a11 such mappings forms a group in which the unit is the identity map. The group is called the

~-symmetric group on n letters. Let us for simplicity denote the set of n abjects by the numbers

1,2,..., n. The mapping S such that S(i) = i + 1 mod n Will be de- noted by (12!3. . .n) and more generally (i j . . . m) Will denote the map- pingTsuchthatT(i)=j,...,T(m)=i.If(ij...m)hasknumbers, then it Will be called a k cycle. It is clear that if T = (i j . . . s) then

T-i = (s.. .ji).

We now establish the

-~-Lemma. If a subgroup U of the symmetric group on n letters

~~(n > 4) contains every 3-cycle, and if u is a normal subgroup of U ~~such that U,/u is abelian, then u contains every 3-cycle. Proof: Let f be the natural homomorphism f(U) = U/u and let x = (ijk), y = (krs) be twoelements of U, where i, j, k, r, s are 5 numbers. Then since V/u is abelian, setting f(x) = x’ , f(y) = y ’ we have f(x-‘y-‘xy) = ~‘-~y’-rx’y’ = 1, so that x-‘y-‘xy E u. But x-‘y-‘xy = (kji).(srk).(ijk).(krs) = (kjs) and for each k, j, s we have (kjs) #F u.

THEOREM-~~ 4. The symmetric group G on n letters is not solvable for n > ,4.

If there were a sequence exhibiting the solvability, since G con- tains every 3-cycle, SO would each succeeding group, and the sequence could not end with the unit. 72

C. Solutïon-- of Equations by Radicals. The extension field E over F is called an extension by radicals if there exist intermediate fields B, , B, , . . . , Br = E and Bi = Bi-r( a i) where each ai is a root of an equation of the form xni - ai = 0, ai E Bi-, . A polynomial f ( x ) in a field F is said to be solvable by radicals if its splitting field lies in an extension by radicals. We assume unless otherwise specified that the base field has characteristic 0 and that F contains as many roots of unity as are needed to make our sub- sequent statements valid.

Let us remark first that any extension of F by radicals cari always be extended to an extension of F by radicals which is normal over F. Indeed B, is a normal extension of B, since it contains not only ar, but cal, where E is any n,-root of unity, SO that B, is the splitting field of xnl - a,. If fr(x) =TT(xn2 - c( a2 )), where 0 takes a11 values in u the group of automorphisms of B, over BO, then f, is in BO, and ad- joining successively the roots of xn2 - o( a 2) brings us to an exten- sion of B2 which is normal over F. Continuing in this way we arrive at an extension of E by radicals which Will be normal over F. We now prove

THEOREM- 5. The polynomial f(x) is solvable by radicals if and only if its group is solvable.

Suppose f(x) is solvable by radicals. Let E be a normal exten- sion of F by radicals containing the splïtting field B of f(x), and cal1 G the group of E over F. Since for each i, Bi is a Kummer extension of Bi_r, the group of Bi over B,-r is abelian. In the sequence of groups 73

G = GB 11 GB 3 . . . 3 GB = 1 each is a normal subgroup of the 1 r precediig since G is the group of E over Bis1 and Bi is a normal Bi-l extension of B,-i. But GB, /GB, is the group of B, over B,-i and hence 1-l 1 is abelian. Thus G is solvable. However, G, is a normal subgroup of G, and G/G, is the group of B over F, and is therefore the group of the polynomial f(x). But G/G, is a homomorph of the solvable group G and hence is itself solvable.

On the other hand, suppose the group G of f(x) to be solvable and let E be the splitting field. Let G = Go 1 G, 1 . . . 1 Gr = 1 be a sequence with abelian factor groups. Cal1 Bi the fixed field for Gi. Since G,-i is the group of E over B,-i and Gi is a normal subgroup of Gi-l, then Bi is normal over B,-i and the group Ci-i/G, is abelian. Thus Bi is a Kummer extension of Bi_i, hence is splitting field of a polynomial

oftheform(x”-a,)(x”-a2)...(xn-as) SO that by forming the successive splitting fields of the x” - ak we see that Bi is an extension of Biml by radicals, from which it follows that E is an extension by radicals.

Remark. The assumption that F contains roots of unity is not necessary in the above theorem. For if f(x) has a solvable group G, then we may adjoin to F a primitive nth root of unity, where n is, say, equal to the order of G. The group of f(x) when considered as lying in F’ is, by the theorem on Natural Irrationalities, a subgroup G’ of G, and hence is solvable. Thus the splitting field over F’ of f(x) cari be obtained by radicals. Conversely, if the splitting field E over F of f(x) cari be obtained by radicals, then by adjoining a suitable root of unity E is extended to E’ which is still normal over F’ . But E’ could be 74 obtained by adjoining first the root of unity, and then the radicals, to

F; F would first be extended to F ’ and then F ’ would be extended to

E ’ . Calling G the group of E ’ over F and G ’ the group of E ’ over F ’ , we see that G ’ is solvable and G/G ’ is the group of F ’ over F and hence abelian. Thus G is solvable. The factor group G/G, is the group of f(x) and being a homomorph of a solvable group is also solvable.

D. The General Equation of Degree n.

If F is a field, the collection of rational expressions in the variables ui, up, . . . , un with coefficients in F is a field F(ui, u2, . . . , un).

By the general equation of degree n we mean the equation

(1) f(x) = x” - ulx”-i + l12x”-2 - + . . . + (-l)“u,. Let E be the splitting field of f(x) over F(ui, II~,. . . , un). If

V l’VZ>...> v, are the roots of f(x) in E, then u1 = v1 + v2 + . . . + vn> u2 = v1v2 + v1v3, + . . . + vnel Vn’. .

. . . , un = VI . v2 . . . . . vn .

We shall prove that the group of E over F(ui, u2, . . . , un) is the symmetric group.

LetF(x,,x,,..., xn) be the field generated from F by the variables xi, x2,. . . ,x,. Let ai = xi + x2 + . . . + x”,

= x1x*. . . a2 = x1x2 + x1x3 + . . . + XnelXn >...> an xn be the ele- mentary symmetric functions, i.e., (x-x ,)(x-x 2). . . (x-x,) =

X” - alx”-’ + - . . .(-l)% ” = f*(x). If g(al,a2, . . .,a,) is a polynomial in ai, . . . ,a,, then g(a, ,a2, . . . ,a,) = 0 only if g is the 75 zero polynomial. For if g( xx,, cxixk, . . .) = 0, then this relation would hold also if the xi were replaced by the vi. Thus, g(cvi,cviv,,...) = Oorg(ul,uz,...,un) = Ofromwhichitfollows that g is identically zero. Between the subfield F(a,, . . . , an) of F( xi, . . . , xn) and F(ul,u2,. . . > un) we set up the following correspondence: Let f(u,, *. . ,11,)/g(u,, . . . >un) be an element of F(ui,. . . ,un). We make this correspond to f(a,, . . . ,a,)/g(a,, . . . ,a,). This is clearly a map- ping of F(ui,uz,. . . ,u,) on a11 of F(ai , . . . ,a,). Moreover, if f(al,a2,...,an)/g(a1,a~,...,a~) = fi (al,az!, . . . ,a,)/gl(al,a2,. . . ,a*), then fg, - gf, = 0. But this implies by the above that f(u,, . ..9U,h$Ul,. . .>U,) - g(u,, . ..,Un)‘f,(U,, . . .>U,) = 0

SO that f(u,, . . . ,u,)/g(u1,u2,. . . ,u,) = fi(U],..., U”)/6cl(UI>U2>...> un). It follows readily from this that themappingof F(u,,u, ,..., un) on F(a,,a,,. ..,an)is an isomor- phism. But under this correspondence f(x) corresponds to f * ( x). SinceE:andF(x,,x,,..., xn) are respectively splitting fields of f(x) and f * I[X), by Theorem 10 the isomorphism cari be extended to an iso- morphism between E and F (xi, x2, . . . , xn). Therefore, the group of E over F(ur,uz,. . . , un) is isomorphic to the group of F ( x1, x2, . . . , xn) over F(al,a2,. . . ,a,). E:ach. permutation of xi, x2,. . . , xn leaves al,az, . . . ,a,, fixed and, therefore, induces an automorphism of F( xi, x2, . . . , x”) which leaves F(a,,a*, . . . , a,.,) fixed. Conversely, each automorphism of

F(xl>xZ>...> x,, ) which leaves F(a 1, . . . , a,, ) fixed must permute the roots xi, x:!, . . . , xn of f*(x) and is completely determined by the 76 permutation it effects on x1, x2, . . . , xn. Thus, the group of F( x1, x2, . . . , xn) over F(a1,a2,. . .,a,)is th e s y mmetric group on n letters. Because of the isomorphism between F ( x1, . . . , xn ) and E, the group for E over

W++. . . >u,,) is also the symmetric group. If we remark that the symmetric group for n > 4 is not solvable, we obtain from the theorem on solvability of equations the famous theorem of Abel:

THEOREM 6. The group of the general equation of degree n is

~~the symmetric group on n letters. The general equation of degree n is not solvable by radicals if n > 4.

E. Solvable Equations of Prime Degree.

The group of an equation cari always be considered as a permu- tation group. If f(x) is a polynomial in a field F, let a,, a2, . . . , c, be the roots of f(x) in the splitting field E = F( ar, . . . , an). Then each automorphism of E over F maps each root of f(x) into a root of f(x), that is, permutes the roots. Since E is generated by the roots of f(x), different automorphisms must effect distinct permutations. Thus, the group of E over F is a permutation group acting on the roots al,a2,...,Qn of f(x). For an irreducible equation this group is always transitive. For

let a and a’ be any two roots of f(x), where f(x) is assumed irreduci- ble. F(a ) and F(a ’ ) are isomorphic where the isomorphism is the

identity on F, and this isomorphism cari be extended to an automorphism

of E (Theorem 10). Thus, there is an automorphism sending any given

root into any other root, which establishes the “transitivity” of the group. 77

A permutation o of the numbers 1,2, , . . , q is called a linear ~~-substitution modulo q if there exists a number b b 0 modulo q such that o(i) :E bi + c(mod q), i = 1,2,. . . ,q.

--THEOREM 7. Let f( x ) be an irreducible equation of prime de- ~-gree q in a field F. The group G of f( x) (which is a permutation group ~~of the roots, or the numbers 1,2, . . . , q) is solvable if and only if, ~~after a suitable change in the numbering of the roots, G is a group of ~~linear substitutions modulo q, and in the group G a11 the substitutions ~-withb = l,o(i) = c + l(c = 1,2 ,..., q)occur. Let G be a transitive substitution group on the numbers 1,2,. . . , q and let G, be a normal subgroup of G. Let 1,2,. . . , k be the images of 1 under the permutations of G,; we say: 1,2, . . . , k is a e-domain of transitivity of G,. If i <- q is a number not belonging to this domain of transitivity, there is a o E G which maps 1 on i. Then 0(1,2,..., k) is a domain of transitivity of oGlu-‘. Since G, is a normal subgroup of G, we have G, = oG,o-‘. Thus, (T( 1,2,. . . , k) is again a domain of transitivity of G, which contains the integer i and has k elements. Since i was arbitrary, the domains of transitivity of G, a11 contain k elements. Thus, the numbers 1,2, . . . , q are divided into a collection of mutually exclusive sets, each containing k ele- ments, SO that k is a divisor of q. Thus, in case q is a prime, either k = 1 (and then G, consists of the unit alone) or k = q and G, is also transitive.

TO prove the theorem, we consider the case in which G is solvable. Let G = G0 7> G, 3 . . . 3 Gs+l = 1 be a sequence exhibiting the solvability. Since G, is abelian, choosing a cyclic subgroup of it 78 would permit us to assume the term before the last to be cyclic, i.e., Gs is cyclic. If ~7 is a generator of Gs, CJ must consist of a cycle con- taining a11 q of the numbers 1,2, . . . , q since in any other case Gs would not be transitive [ if 7~7 -l isanelementofGs,sayrm-1=ob.Let7(i) = jorr-l(j) = i, then ro-r-l( j) = ob( j) = j + b (mod q). Therefore, Ta(i) E r(i) + b (mod q) or r(i+l) = r(i) + b for each i. Thus, setting T(O) = c, we have r(l) = c + b, r(2) = r( 1) + b = c + 2b and in general 7(i) E c + ib (mod q). Thus, each substitution in G s-l is a linear substitution. Moreover, the only elements of Gsml which leave no element fixed belong to Gs, since for each a f 1, there is an i such that ai + b = i (mod q) [ take i such that (a-l) i z - b]. We prove by an induction that the elements of G are a11 linear substitutions, and that the only cycles of q letters belong to Gs. Sup- pose the assertion true of Gsq. Let r c Gsmnml and let v be a cycle which belongs to Gs (hence also to G,-,). Since the transform of a cycle is a cycle, r-107 is a cycle in Gs-, and hence belongs to Gn.

Thus T-~UT = ub for some b. By the argument in the preceding para-

graph, r is a linear substitution bi + c and if 7 itself does not belong to

Gs, then 7 leaves one integer fixed and hence is not a cycle of q elements. 79

We now prove the second half of the theorem. Suppose G is a

group of linear substitutions which contains a subgroup N of the form c(i) 5: i -+ c. Since the only linear substitutions which do not leave an integer fixed belong to N, and since the transform of a cycle of q elements is again a cycle of q elements, N is a normal subgroup of G. In each coset N . r where r(i) = bi + c the substitution 0-l~ occurs, where (T E i + c. But o-ir( i) = (bi + c) - c F bi. Moreover, if

r(i) z biandr’(i) = b’i thenrr’(i) E bb’i. Thus, thefactorgroup (G/N) is isomorphic to a multiplicative subgroup of the numbers 1,2,. . . , q-1 mod q and is therefore abelian. Since (G/N) and N are both abelian, G is solvable.

--~Corollary 1. If G is a solvable transitive substitution group on q . *

~-letters (q prime), then the only substitution of G which leaves two or ~~more letters fixed is the identity. This follows from the fact that each substitution is linear modula

q and bi + c E i (mod q) has either no solution (b z 1, c + 0) or exactly one solution(b f 1) unless b = 1, c = 0 in which case the sub- stitution is the identity.

--~Corollary 2. A solvable, irreducible equation of prime degree in ~~a field which is a subset of the real numbers has either one real root or a11 its roots are real.

The group of the equation is a solvable transitive substitution

group on q (prime) letters. In the splitting field (contained in the field of complex numbers) the automorphism which maps a number into its complex conjugate would leave fixed a11 the real numbers. By Corollary 80

1, if two roots are left fixed, then a11 the roots are left fixed, SO that if the equation has two real roots a11 its roots are real.

F. Ruler and Compass Constructions. Suppose there is given in the plane a finite number of elementary geometric figures, that is, points, straight lines and circles. We seek to construct others which satisfy certain conditions in terms of the given figures. Permissible steps in the construction Will entai1 the choice of an arbitrary point interior to a given region, drawing a line through two points and a circle with given tenter and radius, and finally intersec- ting pairs of lines, or circles, or a line and circle. Since a straight line, or a line segment, or a circle is determined by two points, we cari consider ruler and compass constructions as con- structions of points from given points, subject to certain conditions. If we are given two points we may join them by a line, erect a perpendicular to this line at, say, one of the points and, taking the dis- tance between the two points to be the unit, we cari with the compass lay off any integer n on each of the lines. Moreover, by the usual method, we cari draw parallels and cari construct m/n. Using the two lines as axes of a cartesian coordinate system, we cari with ruler and compass construct a11 points with rational coordinates. Ifa,b,c,... are numbers involved as coordinates of points which determine the figures given, then the sum, product, difference and quotient of any two of these numbers cari be constructed. Thus, each 81 element of the field R( a, b, c, . . .) which they generate out of the rational numbers cari be constructed.

It is required that an arbitrary point is any point of a given region. If a construction by ruler and compass is possible, we cari always choose our arbitrary points as points having rational coordinates. If we join two points with coefficients in R( a, b, c, . . . ) by a line, its equa- tion Will have coefficients in R( a, b, c, . . .) and the intersection of two such lines Will be a point with coordinates in R( a, b, c, . . . ). The equa- tion of a circle Will have coefficients in the field if the circle passes through three points whose coordinates are in the field or if its tenter and one point have coordinates in the field. However, the coordinates of the intersection of two such circles, or a straight line and circle, Will involve square roots.

It follows that if a point cari be constructed with a ruler and com- pass, its coordinates must be obtainable from R( a, b, c, . . . ) by a formula only involving square roots, that is, its coordinates Will lie in a field RS 3 Rs-i 3 . . . 3 R, = R(a,b,c,... ) where each field Ri is splitting field over Ri-r of a quadratic equation x2 - a = 0. It follows (Theorem 6, p. 21) since either Ri = Ri-r or ( Ri/Ri-r ) = 2, that (RJR, ) is a power of two. If x is the coordinate of a constructed point, tben

(Rr( x)/R, ) * ( RS/R, (x)) = (RJR, ) = 2” SO that Rr( x)/R, must also be a power of two.

Conversely, if the coordinates of a point cari be obtained from R(a,b,c,... ) by a formula involving square roots only, then the point cari be constructed by ruler and compass. For, the field operations of 82 addition, subtraction, multiplication and division may be performed by ruler and compass constructions and, also, square roots using 1: r = r : rl to obtain r = d rI may be performed by means of ruler and compass instructions.

As an illustration of these considerations, let us show that it is impossible to trisect an angle of 604 Suppose we have drawn the unit circle with tenter at the vertex of the angle, and set up our coordinate system with X-axis as a side of the angle and origin at the vertex. Trisection of the angle would be equivalent to the construction of the point (COS 20”, sin 209 on the unit circle. From the equation

COS 38 = 4 cos3 0 - 3 COS 8, the abscissa would satisfy 4x3 -. 3x = 1/2. The reader may readily verify that this equation has no rational roots, and is therefore irreducible in the field of rational numbers. But since we may assume only a straight line and unit length given, and since the 60° angle cari be constructed, we may take

R(a,b,c,. . ..) to be the field R of rational numbers. A root a of the irreducible equation 8x3 - 6x - 1 = 0 is such that (R(a)/R) = 3, and not a power of two.